💡Okay, so I have this problem where I need to find the maximum of the function (frac{xy}{x^2 + y^2}) within a specific domain. The domain is given as (frac{1}{3} leq x leq frac{3}{5}) and (frac{1}{4} leq y leq frac{1}{2}). Hmm, okay, let me try to figure this out step by step.First, I remember that when dealing with functions of two variables, especially when looking for maxima or minima, it's useful to consider the critical points inside the domain and also check the boundaries. But before jumping into calculus, maybe there's a way to simplify the function or use some substitution to make it easier.Looking at the function (frac{xy}{x^2 + y^2}), I notice that both the numerator and the denominator are homogeneous functions of degree 2. That means if I scale (x) and (y) by a factor, say (k), the function remains the same. So, maybe I can use a substitution to reduce the number of variables.Let me try setting (t = frac{x}{y}). Then, (x = ty). Substituting this into the function:[frac{xy}{x^2 + y^2} = frac{ty cdot y}{(ty)^2 + y^2} = frac{ty^2}{t^2 y^2 + y^2} = frac{t}{t^2 + 1}]Oh, that's interesting! So, the function simplifies to (frac{t}{t^2 + 1}), where (t = frac{x}{y}). Now, I can analyze this function of a single variable (t) to find its maximum.Let me denote (f(t) = frac{t}{t^2 + 1}). To find its maximum, I can take the derivative and set it equal to zero. The derivative of (f(t)) with respect to (t) is:[f'(t) = frac{(1)(t^2 + 1) - t(2t)}{(t^2 + 1)^2} = frac{t^2 + 1 - 2t^2}{(t^2 + 1)^2} = frac{1 - t^2}{(t^2 + 1)^2}]Setting the derivative equal to zero to find critical points:[1 - t^2 = 0 implies t^2 = 1 implies t = pm 1]Since (x) and (y) are both positive in the given domain (because they are fractions between 0 and 1), (t = frac{x}{y}) will also be positive. So, the critical point we're interested in is (t = 1).Now, let's evaluate (f(t)) at (t = 1):[f(1) = frac{1}{1^2 + 1} = frac{1}{2}]So, the maximum value of (f(t)) is (frac{1}{2}) when (t = 1), which implies (x = y). But wait, in our domain, (x) ranges from (frac{1}{3}) to (frac{3}{5}) and (y) ranges from (frac{1}{4}) to (frac{1}{2}). Is there a point where (x = y) within this domain?Let's see. The overlap of the intervals for (x) and (y) is from (frac{1}{3}) to (frac{1}{2}). So, yes, there are points where (x = y) within this domain. For example, (x = y = frac{1}{2}) is within the domain since (frac{1}{2}) is less than (frac{3}{5}) and greater than (frac{1}{3}), and similarly for (y).But wait, hold on. When (x = y), the function (frac{xy}{x^2 + y^2}) becomes (frac{x^2}{2x^2} = frac{1}{2}). So, the maximum value of the function is (frac{1}{2}) when (x = y). But is this actually the maximum within the given domain?Hmm, maybe I should check the boundaries as well because sometimes the maximum can occur on the boundary of the domain, especially if the critical point is inside the domain but the function could be larger on the edges.So, to be thorough, I should check the function at all four corners of the domain rectangle defined by (x = frac{1}{3}), (x = frac{3}{5}), (y = frac{1}{4}), and (y = frac{1}{2}).Let me list the four corners:1. ((frac{1}{3}, frac{1}{4}))2. ((frac{1}{3}, frac{1}{2}))3. ((frac{3}{5}, frac{1}{4}))4. ((frac{3}{5}, frac{1}{2}))I'll compute the function value at each of these points.1. At ((frac{1}{3}, frac{1}{4})):[frac{(frac{1}{3})(frac{1}{4})}{(frac{1}{3})^2 + (frac{1}{4})^2} = frac{frac{1}{12}}{frac{1}{9} + frac{1}{16}} = frac{frac{1}{12}}{frac{16 + 9}{144}} = frac{frac{1}{12}}{frac{25}{144}} = frac{1}{12} times frac{144}{25} = frac{12}{25} = 0.48]2. At ((frac{1}{3}, frac{1}{2})):[frac{(frac{1}{3})(frac{1}{2})}{(frac{1}{3})^2 + (frac{1}{2})^2} = frac{frac{1}{6}}{frac{1}{9} + frac{1}{4}} = frac{frac{1}{6}}{frac{4 + 9}{36}} = frac{frac{1}{6}}{frac{13}{36}} = frac{1}{6} times frac{36}{13} = frac{6}{13} approx 0.4615]3. At ((frac{3}{5}, frac{1}{4})):[frac{(frac{3}{5})(frac{1}{4})}{(frac{3}{5})^2 + (frac{1}{4})^2} = frac{frac{3}{20}}{frac{9}{25} + frac{1}{16}} = frac{frac{3}{20}}{frac{144 + 25}{400}} = frac{frac{3}{20}}{frac{169}{400}} = frac{3}{20} times frac{400}{169} = frac{60}{169} approx 0.355]4. At ((frac{3}{5}, frac{1}{2})):[frac{(frac{3}{5})(frac{1}{2})}{(frac{3}{5})^2 + (frac{1}{2})^2} = frac{frac{3}{10}}{frac{9}{25} + frac{1}{4}} = frac{frac{3}{10}}{frac{36 + 25}{100}} = frac{frac{3}{10}}{frac{61}{100}} = frac{3}{10} times frac{100}{61} = frac{30}{61} approx 0.4918]So, the function values at the four corners are approximately 0.48, 0.4615, 0.355, and 0.4918. The highest among these is approximately 0.4918, which is (frac{30}{61}).Wait a minute, earlier I thought the maximum was (frac{1}{2}) when (x = y), but in the corners, the maximum is about 0.4918, which is less than (frac{1}{2}). So, does that mean the maximum is actually (frac{1}{2}) somewhere inside the domain?But hold on, when (x = y), the function is (frac{1}{2}), which is indeed higher than the values at the corners. So, is (frac{1}{2}) the maximum?But let me verify if (x = y) is within the domain. Since (x) can be as low as (frac{1}{3}) and as high as (frac{3}{5}), and (y) can be as low as (frac{1}{4}) and as high as (frac{1}{2}), the overlapping region where (x = y) is from (frac{1}{3}) to (frac{1}{2}). So, points like ((frac{1}{2}, frac{1}{2})) are indeed within the domain.But wait, when I plug in (x = y = frac{1}{2}), let's compute the function:[frac{(frac{1}{2})(frac{1}{2})}{(frac{1}{2})^2 + (frac{1}{2})^2} = frac{frac{1}{4}}{frac{1}{4} + frac{1}{4}} = frac{frac{1}{4}}{frac{1}{2}} = frac{1}{2}]So, that's correct. The function reaches (frac{1}{2}) at (x = y = frac{1}{2}). But wait, earlier when I computed the function at the corners, the maximum was (frac{30}{61} approx 0.4918), which is less than (frac{1}{2}). So, that suggests that the maximum is indeed (frac{1}{2}) at the point ((frac{1}{2}, frac{1}{2})).But hold on, let me double-check. Maybe I made a mistake in my earlier substitution or reasoning.I set (t = frac{x}{y}), so the function becomes (frac{t}{t^2 + 1}). The maximum of this function occurs at (t = 1), giving (frac{1}{2}). So, that's correct.But in the domain, (t = frac{x}{y}) can vary. Let's see what the possible range of (t) is.Given (x in [frac{1}{3}, frac{3}{5}]) and (y in [frac{1}{4}, frac{1}{2}]), the minimum value of (t = frac{x}{y}) occurs when (x) is minimum and (y) is maximum, so:[t_{text{min}} = frac{frac{1}{3}}{frac{1}{2}} = frac{2}{3}]And the maximum value of (t) occurs when (x) is maximum and (y) is minimum:[t_{text{max}} = frac{frac{3}{5}}{frac{1}{4}} = frac{12}{5} = 2.4]So, (t) ranges from (frac{2}{3}) to (frac{12}{5}). Now, the function (f(t) = frac{t}{t^2 + 1}) has its maximum at (t = 1), which is within the interval ([frac{2}{3}, frac{12}{5}]). Therefore, the maximum value of (f(t)) is indeed (frac{1}{2}) at (t = 1), which corresponds to (x = y).But wait, earlier when I evaluated the function at the corners, the maximum was (frac{30}{61} approx 0.4918), which is less than (frac{1}{2}). So, that suggests that the maximum is achieved not at the corners but somewhere inside the domain where (x = y).But let me confirm this by checking another point where (x = y) within the domain. For example, let's take (x = y = frac{1}{2}), which we already did, giving (frac{1}{2}). Another point could be (x = y = frac{1}{3}), but wait, (y) can't be (frac{1}{3}) because (y) is only up to (frac{1}{2}). So, the overlapping region is from (frac{1}{3}) to (frac{1}{2}).Wait, actually, (x) can go up to (frac{3}{5}), which is 0.6, and (y) can go up to (frac{1}{2}), which is 0.5. So, (x = y = frac{1}{2}) is within the domain, but (x = y = frac{3}{5}) is not because (y) can't be (frac{3}{5}). So, the maximum (x = y) can be is (frac{1}{2}).Therefore, the function reaches (frac{1}{2}) at ((frac{1}{2}, frac{1}{2})), which is within the domain. So, that should be the maximum.But wait, earlier when I computed the function at the corners, the maximum was (frac{30}{61} approx 0.4918), which is less than (frac{1}{2}). So, that suggests that the maximum is indeed (frac{1}{2}) at ((frac{1}{2}, frac{1}{2})).But I'm a bit confused because when I substituted (t = frac{x}{y}), I found that the maximum of (f(t)) is (frac{1}{2}) at (t = 1), which corresponds to (x = y). So, that should be the maximum.Wait, but let me think again. The function (frac{xy}{x^2 + y^2}) can also be written as (frac{1}{frac{x}{y} + frac{y}{x}}), which is the same as (frac{1}{t + frac{1}{t}}). So, to maximize this, we need to minimize (t + frac{1}{t}).The function (g(t) = t + frac{1}{t}) has its minimum at (t = 1), where (g(1) = 2). So, the maximum of (frac{1}{g(t)}) is (frac{1}{2}).Therefore, regardless of the domain, as long as (t = 1) is within the possible range of (t), the maximum of the function is (frac{1}{2}).But in our case, (t) ranges from (frac{2}{3}) to (frac{12}{5}), and (t = 1) is within this interval. Therefore, the maximum value of the function is indeed (frac{1}{2}).Wait, but earlier when I evaluated the function at the corners, the maximum was (frac{30}{61} approx 0.4918), which is less than (frac{1}{2}). So, that suggests that the function reaches its maximum not at the corners but at the point where (x = y = frac{1}{2}).Therefore, the maximum value of the function is (frac{1}{2}).But hold on, let me double-check my calculations for the corners because I might have made a mistake.1. At ((frac{1}{3}, frac{1}{4})):[frac{(frac{1}{3})(frac{1}{4})}{(frac{1}{3})^2 + (frac{1}{4})^2} = frac{frac{1}{12}}{frac{1}{9} + frac{1}{16}} = frac{frac{1}{12}}{frac{16 + 9}{144}} = frac{frac{1}{12}}{frac{25}{144}} = frac{1}{12} times frac{144}{25} = frac{12}{25} = 0.48]That seems correct.2. At ((frac{1}{3}, frac{1}{2})):[frac{(frac{1}{3})(frac{1}{2})}{(frac{1}{3})^2 + (frac{1}{2})^2} = frac{frac{1}{6}}{frac{1}{9} + frac{1}{4}} = frac{frac{1}{6}}{frac{4 + 9}{36}} = frac{frac{1}{6}}{frac{13}{36}} = frac{1}{6} times frac{36}{13} = frac{6}{13} approx 0.4615]That's correct too.3. At ((frac{3}{5}, frac{1}{4})):[frac{(frac{3}{5})(frac{1}{4})}{(frac{3}{5})^2 + (frac{1}{4})^2} = frac{frac{3}{20}}{frac{9}{25} + frac{1}{16}} = frac{frac{3}{20}}{frac{144 + 25}{400}} = frac{frac{3}{20}}{frac{169}{400}} = frac{3}{20} times frac{400}{169} = frac{60}{169} approx 0.355]Correct.4. At ((frac{3}{5}, frac{1}{2})):[frac{(frac{3}{5})(frac{1}{2})}{(frac{3}{5})^2 + (frac{1}{2})^2} = frac{frac{3}{10}}{frac{9}{25} + frac{1}{4}} = frac{frac{3}{10}}{frac{36 + 25}{100}} = frac{frac{3}{10}}{frac{61}{100}} = frac{3}{10} times frac{100}{61} = frac{30}{61} approx 0.4918]That's correct as well.So, the function reaches its maximum at the point ((frac{1}{2}, frac{1}{2})) with a value of (frac{1}{2}), which is higher than the values at the corners. Therefore, the maximum value of the function in the given domain is (frac{1}{2}).But wait, earlier I thought the maximum was (frac{6}{13}) based on the substitution, but that seems contradictory. Let me re-examine that.When I set (t = frac{x}{y}), I transformed the function into (frac{t}{t^2 + 1}). The maximum of this function is indeed (frac{1}{2}) at (t = 1). However, in the given domain, (t) can vary from (frac{2}{3}) to (frac{12}{5}). So, the function (f(t) = frac{t}{t^2 + 1}) is increasing on ((0,1)) and decreasing on ((1, infty)). Therefore, within the interval ([frac{2}{3}, frac{12}{5}]), the maximum occurs at (t = 1), giving (frac{1}{2}).So, that confirms that the maximum value is (frac{1}{2}) at (t = 1), which corresponds to (x = y). Therefore, the maximum value of the function is (frac{1}{2}).But wait, earlier when I computed the function at the point ((frac{1}{2}, frac{1}{2})), I got (frac{1}{2}), which is indeed the maximum. So, why did I initially think the maximum was (frac{6}{13})? Maybe I confused it with another problem.Wait, no, in the initial problem, the user wrote:"We can write[frac{xy}{x^2 + y^2} = frac{1}{frac{x}{y} + frac{y}{x}}.]Let (t = frac{x}{y}), so (frac{x}{y} + frac{y}{x} = t + frac{1}{t}). We want to minimize this denominator to maximize (frac{xy}{x^2 + y^2}).Using the function [f(t) = t + frac{1}{t},]we analyze the behavior for different intervals of (t). The function is decreasing in the interval ((0,1]) and increasing in the interval ([1,infty)). Thus, we seek the extreme values of (t) to find the minimum of (f(t)).The minimum of (t) occurs when (x) is minimum and (y) is maximum, i.e., (x = frac{1}{3}) and (y = frac{1}{2}), giving[t = frac{x}{y} = frac{1}{3} div frac{1}{2} = frac{2}{3}.]The maximum of (t) occurs when (x) is maximum and (y) is minimum, i.e., (x = frac{3}{5}) and (y = frac{1}{4}), giving[t = frac{x}{y} = frac{3}{5} div frac{1}{4} = frac{12}{5}.]Calculating the function values at these (t) values:[fleft(frac{2}{3}right) = frac{2}{3} + frac{3}{2} = frac{4}{3} + frac{9}{6} = frac{13}{6},][fleft(frac{12}{5}right) = frac{12}{5} + frac{5}{12} = frac{144}{60} + frac{25}{60} = frac{169}{60}.]The minimum value of (f(t)) is (frac{13}{6}), which occurs at (t = frac{2}{3}). Thus, the maximum of (frac{xy}{x^2 + y^2}) is[frac{1}{fleft(frac{2}{3}right)} = frac{1}{frac{13}{6}} = frac{6}{13}.]Conclusion: The maximum value of (frac{xy}{x^2 + y^2}) in the given domain is (boxed{frac{6}{13}})."Wait, so in this initial solution, the user found that the maximum is (frac{6}{13}), but in my analysis, I found that the maximum is (frac{1}{2}). There must be a mistake here.Let me re-examine the initial substitution. The user wrote:[frac{xy}{x^2 + y^2} = frac{1}{frac{x}{y} + frac{y}{x}}]Yes, that's correct because:[frac{xy}{x^2 + y^2} = frac{1}{frac{x}{y} + frac{y}{x}}]So, to maximize (frac{xy}{x^2 + y^2}), we need to minimize (frac{x}{y} + frac{y}{x}).The function (f(t) = t + frac{1}{t}) has its minimum at (t = 1), where (f(1) = 2). Therefore, the maximum of (frac{xy}{x^2 + y^2}) is (frac{1}{2}).But in the initial solution, the user considered the extreme values of (t), which are (frac{2}{3}) and (frac{12}{5}), and found that (f(t)) is minimized at (t = frac{2}{3}), giving (frac{13}{6}), and thus the maximum of the original function is (frac{6}{13}).But this contradicts my earlier conclusion that the maximum is (frac{1}{2}). So, where is the mistake?Wait, the function (f(t) = t + frac{1}{t}) is minimized at (t = 1), but in the given domain, (t) can only vary between (frac{2}{3}) and (frac{12}{5}). So, the minimum of (f(t)) in this interval is not necessarily at (t = 1), but we need to check the behavior of (f(t)) within this interval.Wait, (f(t) = t + frac{1}{t}) is decreasing on ((0,1)) and increasing on ((1, infty)). Therefore, in the interval ([frac{2}{3}, frac{12}{5}]), the function (f(t)) will decrease from (t = frac{2}{3}) to (t = 1), and then increase from (t = 1) to (t = frac{12}{5}).Therefore, the minimum of (f(t)) in this interval occurs at (t = 1), giving (f(1) = 2). Therefore, the maximum of (frac{xy}{x^2 + y^2}) is (frac{1}{2}).But in the initial solution, the user found that the minimum of (f(t)) is at (t = frac{2}{3}), giving (frac{13}{6}), and thus the maximum of the original function is (frac{6}{13}). That seems incorrect because the minimum of (f(t)) in the interval is actually at (t = 1), not at (t = frac{2}{3}).Wait, let me compute (f(t)) at (t = frac{2}{3}) and (t = frac{12}{5}):At (t = frac{2}{3}):[fleft(frac{2}{3}right) = frac{2}{3} + frac{3}{2} = frac{4}{6} + frac{9}{6} = frac{13}{6} approx 2.1667]At (t = frac{12}{5}):[fleft(frac{12}{5}right) = frac{12}{5} + frac{5}{12} = frac{144}{60} + frac{25}{60} = frac{169}{60} approx 2.8167]At (t = 1):[f(1) = 1 + 1 = 2]So, indeed, the minimum of (f(t)) in the interval ([frac{2}{3}, frac{12}{5}]) is at (t = 1), giving (f(1) = 2). Therefore, the maximum of (frac{xy}{x^2 + y^2}) is (frac{1}{2}).Therefore, the initial solution was incorrect because it assumed that the minimum of (f(t)) occurs at (t = frac{2}{3}), but in reality, the minimum occurs at (t = 1), which is within the interval.So, the correct maximum value of the function is (frac{1}{2}) at the point ((frac{1}{2}, frac{1}{2})).But wait, in the initial problem statement, the user concluded that the maximum is (frac{6}{13}), which is approximately 0.4615, which is less than (frac{1}{2}). So, that suggests that the initial solution was incorrect.Therefore, the correct maximum value is (frac{1}{2}).But let me double-check by considering another approach. Maybe using partial derivatives to find critical points.Let me denote the function as (f(x, y) = frac{xy}{x^2 + y^2}).To find critical points, I'll compute the partial derivatives and set them equal to zero.First, compute (frac{partial f}{partial x}):[frac{partial f}{partial x} = frac{y(x^2 + y^2) - xy(2x)}{(x^2 + y^2)^2} = frac{y(x^2 + y^2) - 2x^2 y}{(x^2 + y^2)^2} = frac{y(x^2 + y^2 - 2x^2)}{(x^2 + y^2)^2} = frac{y(y^2 - x^2)}{(x^2 + y^2)^2}]Similarly, compute (frac{partial f}{partial y}):[frac{partial f}{partial y} = frac{x(x^2 + y^2) - xy(2y)}{(x^2 + y^2)^2} = frac{x(x^2 + y^2) - 2y^2 x}{(x^2 + y^2)^2} = frac{x(x^2 + y^2 - 2y^2)}{(x^2 + y^2)^2} = frac{x(x^2 - y^2)}{(x^2 + y^2)^2}]Set both partial derivatives equal to zero:1. (frac{partial f}{partial x} = 0 implies y(y^2 - x^2) = 0)2. (frac{partial f}{partial y} = 0 implies x(x^2 - y^2) = 0)From equation 1: Either (y = 0) or (y^2 = x^2). But (y) cannot be zero in the given domain, so (y^2 = x^2 implies y = pm x). Since (x) and (y) are positive, (y = x).From equation 2: Either (x = 0) or (x^2 = y^2). Similarly, (x) cannot be zero, so (x^2 = y^2 implies x = y).Therefore, the only critical point occurs when (x = y). So, within the domain, the critical points are along the line (x = y).Now, to find if this critical point is a maximum, minimum, or saddle point, we can use the second derivative test or analyze the behavior.But since we're looking for the maximum, and we've already found that at (x = y = frac{1}{2}), the function reaches (frac{1}{2}), which is higher than the values at the corners, it's likely that this is the maximum.Therefore, the maximum value of the function is (frac{1}{2}) at the point ((frac{1}{2}, frac{1}{2})).But wait, let me check if there are any other critical points on the boundaries. Sometimes, the maximum can occur on the boundary even if there's a critical point inside.So, let's check the boundaries:1. Boundary where (x = frac{1}{3}): (y) varies from (frac{1}{4}) to (frac{1}{2}).2. Boundary where (x = frac{3}{5}): (y) varies from (frac{1}{4}) to (frac{1}{2}).3. Boundary where (y = frac{1}{4}): (x) varies from (frac{1}{3}) to (frac{3}{5}).4. Boundary where (y = frac{1}{2}): (x) varies from (frac{1}{3}) to (frac{3}{5}).Let's analyze each boundary.1. Boundary (x = frac{1}{3}):The function becomes (f(frac{1}{3}, y) = frac{frac{1}{3} y}{(frac{1}{3})^2 + y^2} = frac{frac{y}{3}}{frac{1}{9} + y^2}).Let me denote this as (f(y) = frac{frac{y}{3}}{frac{1}{9} + y^2}).To find extrema, take the derivative with respect to (y):[f'(y) = frac{frac{1}{3}(frac{1}{9} + y^2) - frac{y}{3}(2y)}{(frac{1}{9} + y^2)^2} = frac{frac{1}{27} + frac{y^2}{3} - frac{2y^2}{3}}{(frac{1}{9} + y^2)^2} = frac{frac{1}{27} - frac{y^2}{3}}{(frac{1}{9} + y^2)^2}]Set (f'(y) = 0):[frac{1}{27} - frac{y^2}{3} = 0 implies frac{1}{27} = frac{y^2}{3} implies y^2 = frac{1}{9} implies y = frac{1}{3}]But (y) is in ([frac{1}{4}, frac{1}{2}]), and (frac{1}{3} approx 0.333) is within this interval. So, we have a critical point at (y = frac{1}{3}).Compute (f(frac{1}{3}, frac{1}{3})):[fleft(frac{1}{3}, frac{1}{3}right) = frac{frac{1}{3} cdot frac{1}{3}}{(frac{1}{3})^2 + (frac{1}{3})^2} = frac{frac{1}{9}}{frac{2}{9}} = frac{1}{2}]So, on this boundary, the function reaches (frac{1}{2}) at (y = frac{1}{3}).2. Boundary (x = frac{3}{5}):The function becomes (f(frac{3}{5}, y) = frac{frac{3}{5} y}{(frac{3}{5})^2 + y^2} = frac{frac{3y}{5}}{frac{9}{25} + y^2}).Let me denote this as (f(y) = frac{frac{3y}{5}}{frac{9}{25} + y^2}).Take the derivative with respect to (y):[f'(y) = frac{frac{3}{5}(frac{9}{25} + y^2) - frac{3y}{5}(2y)}{(frac{9}{25} + y^2)^2} = frac{frac{27}{125} + frac{3y^2}{5} - frac{6y^2}{5}}{(frac{9}{25} + y^2)^2} = frac{frac{27}{125} - frac{3y^2}{5}}{(frac{9}{25} + y^2)^2}]Set (f'(y) = 0):[frac{27}{125} - frac{3y^2}{5} = 0 implies frac{27}{125} = frac{3y^2}{5} implies y^2 = frac{27}{125} cdot frac{5}{3} = frac{9}{25} implies y = frac{3}{5}]But (y) is in ([frac{1}{4}, frac{1}{2}]), and (frac{3}{5} = 0.6) is outside this interval. Therefore, the maximum on this boundary occurs at the endpoints.Compute (f(frac{3}{5}, frac{1}{4})) and (f(frac{3}{5}, frac{1}{2})):We already computed these earlier:- (f(frac{3}{5}, frac{1}{4}) = frac{60}{169} approx 0.355)- (f(frac{3}{5}, frac{1}{2}) = frac{30}{61} approx 0.4918)So, the maximum on this boundary is approximately 0.4918.3. Boundary (y = frac{1}{4}):The function becomes (f(x, frac{1}{4}) = frac{x cdot frac{1}{4}}{x^2 + (frac{1}{4})^2} = frac{frac{x}{4}}{x^2 + frac{1}{16}}).Let me denote this as (f(x) = frac{frac{x}{4}}{x^2 + frac{1}{16}}).Take the derivative with respect to (x):[f'(x) = frac{frac{1}{4}(x^2 + frac{1}{16}) - frac{x}{4}(2x)}{(x^2 + frac{1}{16})^2} = frac{frac{x^2}{4} + frac{1}{64} - frac{2x^2}{4}}{(x^2 + frac{1}{16})^2} = frac{-frac{x^2}{4} + frac{1}{64}}{(x^2 + frac{1}{16})^2}]Set (f'(x) = 0):[-frac{x^2}{4} + frac{1}{64} = 0 implies frac{x^2}{4} = frac{1}{64} implies x^2 = frac{1}{16} implies x = frac{1}{4}]But (x) is in ([frac{1}{3}, frac{3}{5}]), and (frac{1}{4} = 0.25) is less than (frac{1}{3} approx 0.333). Therefore, the critical point is outside the interval. Thus, the maximum on this boundary occurs at the endpoints.Compute (f(frac{1}{3}, frac{1}{4})) and (f(frac{3}{5}, frac{1}{4})):We already computed these earlier:- (f(frac{1}{3}, frac{1}{4}) = frac{12}{25} = 0.48)- (f(frac{3}{5}, frac{1}{4}) = frac{60}{169} approx 0.355)So, the maximum on this boundary is 0.48.4. Boundary (y = frac{1}{2}):The function becomes (f(x, frac{1}{2}) = frac{x cdot frac{1}{2}}{x^2 + (frac{1}{2})^2} = frac{frac{x}{2}}{x^2 + frac{1}{4}}).Let me denote this as (f(x) = frac{frac{x}{2}}{x^2 + frac{1}{4}}).Take the derivative with respect to (x):[f'(x) = frac{frac{1}{2}(x^2 + frac{1}{4}) - frac{x}{2}(2x)}{(x^2 + frac{1}{4})^2} = frac{frac{x^2}{2} + frac{1}{8} - x^2}{(x^2 + frac{1}{4})^2} = frac{-frac{x^2}{2} + frac{1}{8}}{(x^2 + frac{1}{4})^2}]Set (f'(x) = 0):[-frac{x^2}{2} + frac{1}{8} = 0 implies frac{x^2}{2} = frac{1}{8} implies x^2 = frac{1}{4} implies x = frac{1}{2}]Since (x) is in ([frac{1}{3}, frac{3}{5}]), and (frac{1}{2} = 0.5) is within this interval, we have a critical point at (x = frac{1}{2}).Compute (f(frac{1}{2}, frac{1}{2})):[fleft(frac{1}{2}, frac{1}{2}right) = frac{frac{1}{2} cdot frac{1}{2}}{(frac{1}{2})^2 + (frac{1}{2})^2} = frac{frac{1}{4}}{frac{1}{4} + frac{1}{4}} = frac{frac{1}{4}}{frac{1}{2}} = frac{1}{2}]So, on this boundary, the function reaches (frac{1}{2}) at (x = frac{1}{2}).Therefore, after checking all boundaries, the maximum value of the function is (frac{1}{2}), achieved at the points ((frac{1}{3}, frac{1}{3})) and ((frac{1}{2}, frac{1}{2})). However, ((frac{1}{3}, frac{1}{3})) is within the domain since (x = frac{1}{3}) and (y = frac{1}{3}) are both within their respective intervals.Wait, but earlier when I computed the function at ((frac{1}{3}, frac{1}{3})), I got (frac{1}{2}), which is correct. So, the function reaches its maximum at both ((frac{1}{3}, frac{1}{3})) and ((frac{1}{2}, frac{1}{2})).But in the initial solution, the user concluded that the maximum is (frac{6}{13}), which is approximately 0.4615, but we've found that the function actually reaches (frac{1}{2}), which is higher. Therefore, the initial solution was incorrect.In conclusion, after analyzing the function using substitution, partial derivatives, and checking boundaries, the maximum value of (frac{xy}{x^2 + y^2}) in the given domain is (frac{1}{2}), achieved at the points ((frac{1}{3}, frac{1}{3})) and ((frac{1}{2}, frac{1}{2})).

💡Alright, so I've got this problem about a circle and some lines, and I need to find their polar equations and then figure out the ratio of the lengths of two arcs. Hmm, okay, let's take it step by step.First, the problem gives me parametric equations for a circle C. The equations are:x = 5 + 2 cos αy = √3 + 2 sin αWhere α is the parameter. I remember that parametric equations of circles usually have the form x = h + r cos θ and y = k + r sin θ, where (h, k) is the center and r is the radius. So in this case, the center of the circle should be at (5, √3) and the radius is 2. That makes sense.Now, part (1) asks for the polar equation of circle C. Polar coordinates use ρ and θ, where ρ is the distance from the origin and θ is the angle from the positive x-axis. I need to convert the parametric equations into a polar equation.I recall that in polar coordinates, x = ρ cos θ and y = ρ sin θ. So, substituting these into the parametric equations:ρ cos θ = 5 + 2 cos αρ sin θ = √3 + 2 sin αHmm, but α is the parameter here, so I need to eliminate α to get an equation in terms of ρ and θ. Maybe I can square both equations and add them together because cos² α + sin² α = 1.So, let's square both equations:(ρ cos θ - 5)² = 4 cos² α(ρ sin θ - √3)² = 4 sin² αAdding these together:(ρ cos θ - 5)² + (ρ sin θ - √3)² = 4 (cos² α + sin² α) = 4Expanding the left side:(ρ² cos² θ - 10 ρ cos θ + 25) + (ρ² sin² θ - 2√3 ρ sin θ + 3) = 4Combine like terms:ρ² (cos² θ + sin² θ) - 10 ρ cos θ - 2√3 ρ sin θ + 28 = 4Since cos² θ + sin² θ = 1:ρ² - 10 ρ cos θ - 2√3 ρ sin θ + 28 = 4Subtract 4 from both sides:ρ² - 10 ρ cos θ - 2√3 ρ sin θ + 24 = 0So, that's the polar equation of circle C. I think that's correct.Next, part (1) also asks for the Cartesian equation of line l₂. The problem says that l₁ has the polar equation ρ cos θ - √3 ρ sin θ = 3. Then, moving l₁ 3 units to the left gives l₂.First, let me find the Cartesian equation of l₁. Since in polar coordinates, ρ cos θ = x and ρ sin θ = y, so substituting these:x - √3 y = 3That's the Cartesian equation of l₁. Now, moving a line 3 units to the left means shifting it in the negative x-direction. In Cartesian coordinates, shifting a line left by 3 units replaces x with x + 3.So, starting with x - √3 y = 3, replacing x with x + 3:(x + 3) - √3 y = 3Simplify:x + 3 - √3 y = 3Subtract 3 from both sides:x - √3 y = 0So, the Cartesian equation of l₂ is x - √3 y = 0, which can also be written as y = (1/√3) x or y = (√3/3) x.Wait, let me double-check that. If I have x - √3 y = 0, then solving for y gives y = (1/√3) x, which is the same as y = (√3/3) x. Yeah, that's correct.So, part (1) is done. The polar equation of circle C is ρ² - 10 ρ cos θ - 2√3 ρ sin θ + 24 = 0, and the Cartesian equation of l₂ is y = (√3/3) x.Moving on to part (2). Line l₂ intersects circle C at points A and B. I need to find the ratio of the length of the major arc AB to the length of the minor arc AB.First, let me visualize this. We have circle C with center at (5, √3) and radius 2. Line l₂ is y = (√3/3) x, which is a straight line passing through the origin with a slope of √3/3, which is 30 degrees from the x-axis.To find the points of intersection A and B, I need to solve the system of equations:1. The Cartesian equation of circle C: (x - 5)² + (y - √3)² = 42. The equation of line l₂: y = (√3/3) xSubstitute y from equation 2 into equation 1:(x - 5)² + ((√3/3) x - √3)² = 4Let me expand this step by step.First, expand (x - 5)²:(x - 5)² = x² - 10x + 25Next, expand ((√3/3) x - √3)²:Let me factor out √3:√3 ( (1/3) x - 1 )So, squaring this:(√3)² * ( (1/3 x - 1) )² = 3 * ( (1/3 x - 1) )²Now, expand (1/3 x - 1)²:= (1/3 x)² - 2*(1/3 x)*1 + 1²= (1/9)x² - (2/3)x + 1Multiply by 3:3*(1/9 x² - 2/3 x + 1) = (1/3)x² - 2x + 3So, putting it all together, the equation becomes:(x² - 10x + 25) + ( (1/3)x² - 2x + 3 ) = 4Combine like terms:x² + (1/3)x² -10x -2x +25 +3 = 4Convert x² terms:(1 + 1/3)x² = (4/3)x²Combine x terms:-10x -2x = -12xConstants:25 + 3 = 28So the equation is:(4/3)x² -12x +28 = 4Subtract 4 from both sides:(4/3)x² -12x +24 = 0Multiply both sides by 3 to eliminate the fraction:4x² -36x +72 = 0Simplify by dividing all terms by 4:x² -9x +18 = 0Factor the quadratic:x² -9x +18 = (x - 3)(x - 6) = 0So, x = 3 or x = 6Now, find the corresponding y values using y = (√3/3)x:For x = 3:y = (√3/3)*3 = √3So, point A is (3, √3)For x = 6:y = (√3/3)*6 = 2√3So, point B is (6, 2√3)Wait, but let me check if these points are on the circle.For point A (3, √3):(x -5)² + (y -√3)² = (3 -5)² + (√3 -√3)² = (-2)² + 0 = 4 + 0 = 4, which is correct.For point B (6, 2√3):(x -5)² + (y -√3)² = (6 -5)² + (2√3 -√3)² = (1)² + (√3)² = 1 + 3 = 4, which is also correct.So, points A and B are (3, √3) and (6, 2√3).Now, I need to find the ratio of the length of the major arc AB to the minor arc AB.First, let's find the central angle corresponding to arc AB. Since the circle has radius 2, the circumference is 2π*2 = 4π. The length of an arc is given by rθ, where θ is the central angle in radians.But to find θ, I can use the coordinates of A and B. Alternatively, since I know the coordinates, I can find the angle each point makes with the center.Wait, the center of the circle is at (5, √3). So, points A and B are (3, √3) and (6, 2√3). Let me find the vectors from the center to A and B.Vector CA: from (5, √3) to (3, √3) is (-2, 0)Vector CB: from (5, √3) to (6, 2√3) is (1, √3)Now, to find the angle between vectors CA and CB, which is the central angle for arc AB.The formula for the angle between two vectors u and v is:cos θ = (u · v) / (|u| |v|)Compute the dot product:u = (-2, 0), v = (1, √3)u · v = (-2)(1) + (0)(√3) = -2 + 0 = -2|u| = sqrt((-2)^2 + 0^2) = sqrt(4) = 2|v| = sqrt(1^2 + (√3)^2) = sqrt(1 + 3) = sqrt(4) = 2So,cos θ = (-2)/(2*2) = -2/4 = -1/2Therefore, θ = arccos(-1/2) = 2π/3 radians.So, the central angle is 2π/3, which is 120 degrees. That means the minor arc AB has length rθ = 2*(2π/3) = 4π/3.The major arc AB would then be the remaining part of the circle. Since the total circumference is 4π, the major arc length is 4π - 4π/3 = 8π/3.Wait, hold on. The circumference is 2πr = 2π*2 = 4π. So, minor arc is 4π/3, major arc is 4π - 4π/3 = 8π/3.Therefore, the ratio of major arc AB to minor arc AB is (8π/3) : (4π/3) = 2:1.So, the ratio is 2:1.Wait, let me double-check. The central angle is 2π/3, which is less than π, so the minor arc is indeed 2π/3, and the major arc is 2π - 2π/3 = 4π/3. Wait, hold on, no, the circumference is 4π, so the major arc would be 4π - 4π/3 = 8π/3? Wait, no, circumference is 4π, so 4π - 4π/3 = 8π/3, which is correct.But wait, the central angle is 2π/3, so the minor arc length is 2*(2π/3) = 4π/3, and the major arc is 4π - 4π/3 = 8π/3. So, the ratio is 8π/3 : 4π/3 = 2:1.Yes, that seems correct.So, the ratio is 2:1.But let me think again. If the central angle is 2π/3, which is 120 degrees, then the minor arc is 120 degrees, and the major arc is 240 degrees. The ratio of major to minor is 240:120 = 2:1. Yep, that's consistent.So, the ratio is 2:1.I think that's the answer.**Final Answer**(1) The polar equation of circle ( C ) is (boxed{rho^2 - 10rho costheta - 2sqrt{3}rho sintheta + 24 = 0}) and the Cartesian equation of line ( l_2 ) is (boxed{y = frac{sqrt{3}}{3}x}).(2) The ratio of the length of the major arc (widehat{AB}) to the length of the minor arc (widehat{AB}) is (boxed{2}).

💡First, I identify the coordinates of points ( A ) and ( B ). Point ( A ) is at ( (1, -1) ) and point ( B ) is at ( (1, 2) ).To find the vector ( overrightarrow{AB} ), I need to subtract the coordinates of point ( A ) from the coordinates of point ( B ).For the x-component:[x_B - x_A = 1 - 1 = 0]For the y-component:[y_B - y_A = 2 - (-1) = 2 + 1 = 3]Therefore, the vector ( overrightarrow{AB} ) is ( (0, 3) ).

💡First, I need to determine the total number of blisters Brenda has on her arms. Since she has 60 blisters on each arm, I can calculate this by multiplying 60 by 2, which gives me 120 blisters on both arms combined.Next, I know that Brenda has a total of 200 blisters on her entire body. To find out how many blisters are on the rest of her body, I subtract the number of blisters on her arms from the total number of blisters. So, I subtract 120 from 200, which equals 80.Therefore, Brenda has 80 blisters on the rest of her body.

💡Okay, so I have this problem here: In the expansion of (left(x - frac{1}{x^2}right)^n), the binomial coefficients of the 3rd and 8th terms are equal. I need to find the constant term in the expansion. Hmm, okay, let me try to break this down step by step.First, I remember that in the binomial expansion of ((a + b)^n), the general term is given by (T_{k+1} = binom{n}{k} a^{n-k} b^k). So, in this case, (a = x) and (b = -frac{1}{x^2}). That means the general term will be (T_{k+1} = binom{n}{k} x^{n - k} left(-frac{1}{x^2}right)^k).Let me write that out more clearly:[T_{k+1} = binom{n}{k} x^{n - k} cdot (-1)^k cdot x^{-2k} = binom{n}{k} (-1)^k x^{n - 3k}]Okay, so each term is (binom{n}{k} (-1)^k x^{n - 3k}). Now, the problem says that the binomial coefficients of the 3rd and 8th terms are equal. Let me figure out what that means.The 3rd term corresponds to (k = 2) because the first term is (k = 0). Similarly, the 8th term corresponds to (k = 7). So, the binomial coefficients for these terms are (binom{n}{2}) and (binom{n}{7}), respectively. The problem states that these are equal:[binom{n}{2} = binom{n}{7}]I remember that (binom{n}{k} = binom{n}{n - k}). So, if (binom{n}{2} = binom{n}{7}), that implies that either (2 = 7) or (2 = n - 7). Since (2) is not equal to (7), the other possibility is (2 = n - 7). Solving for (n):[2 = n - 7 implies n = 9]So, (n = 9). That’s a key piece of information. Now, I need to find the constant term in the expansion of (left(x - frac{1}{x^2}right)^9).From the general term I wrote earlier:[T_{k+1} = binom{9}{k} (-1)^k x^{9 - 3k}]A constant term means the exponent of (x) is zero. So, I need to find (k) such that:[9 - 3k = 0 implies 3k = 9 implies k = 3]So, the constant term is the 4th term ((k = 3)) in the expansion. Let me compute that:[T_4 = binom{9}{3} (-1)^3 x^{9 - 9} = binom{9}{3} (-1)^3 x^0]Simplifying this:First, compute (binom{9}{3}):[binom{9}{3} = frac{9!}{3!(9 - 3)!} = frac{9 times 8 times 7}{3 times 2 times 1} = 84]Then, ((-1)^3 = -1), and (x^0 = 1). So, putting it all together:[T_4 = 84 times (-1) times 1 = -84]Therefore, the constant term is (-84). Let me just double-check my steps to make sure I didn't make a mistake.1. I identified the general term correctly.2. I set the binomial coefficients equal for the 3rd and 8th terms, which led me to (n = 9).3. Then, I found the value of (k) that makes the exponent zero, which was (k = 3).4. Finally, I calculated the term for (k = 3) and got (-84).Everything seems to check out. I think that's the correct answer.**Final Answer**The constant term in the expansion is boxed{-84}.

💡First, I need to simplify the vector expression (overrightarrow{AC} - overrightarrow{AB} + overrightarrow{CB}).I can start by expressing (overrightarrow{AC}) in terms of (overrightarrow{AB}) and (overrightarrow{BC}) using the triangle rule:[overrightarrow{AC} = overrightarrow{AB} + overrightarrow{BC}]Substituting this back into the original expression:[overrightarrow{AC} - overrightarrow{AB} + overrightarrow{CB} = (overrightarrow{AB} + overrightarrow{BC}) - overrightarrow{AB} + overrightarrow{CB}]Simplifying the expression by canceling out (overrightarrow{AB}):[= overrightarrow{BC} + overrightarrow{CB}]Since (overrightarrow{BC}) and (overrightarrow{CB}) are equal in magnitude but opposite in direction, their sum is the zero vector:[overrightarrow{BC} + overrightarrow{CB} = overrightarrow{0}]Therefore, the simplified expression equals the zero vector.