💡Okay, so I've got this problem here about a function, right? The function is f(x) = 4 sin(π/3 x + π/4). The question is asking about the minimum value of |x₁ - x₂| where x₁ and x₂ are points such that f(x₁) is the minimum value of the function and f(x₂) is the maximum value of the function. And they give me options: A is 6, B is 3, C is 8, D is 2.Alright, let's break this down. First, I remember that the sine function oscillates between -1 and 1. So, when we have 4 sin(something), that means the function will oscillate between -4 and 4. So, f(x) ranges from -4 to 4. Therefore, the minimum value of f(x) is -4, and the maximum is 4.So, x₁ is a point where f(x₁) = -4, and x₂ is a point where f(x₂) = 4. The question is asking for the minimum distance between such x₁ and x₂.Now, to find x₁ and x₂, I need to solve for x when f(x) is at its minimum and maximum. Let's start with the maximum.For f(x) to be maximum, sin(π/3 x + π/4) needs to be 1. So, sin(θ) = 1 when θ = π/2 + 2πk, where k is any integer. So, setting π/3 x + π/4 equal to π/2 + 2πk:π/3 x + π/4 = π/2 + 2πkLet me solve for x:π/3 x = π/2 - π/4 + 2πkπ/3 x = π/4 + 2πkx = (π/4 + 2πk) * (3/π)x = (3/π)(π/4 + 2πk)x = 3/4 + 6kSimilarly, for the minimum value, sin(θ) = -1, which occurs when θ = 3π/2 + 2πk. So,π/3 x + π/4 = 3π/2 + 2πkSolving for x:π/3 x = 3π/2 - π/4 + 2πkπ/3 x = (6π/4 - π/4) + 2πkπ/3 x = 5π/4 + 2πkx = (5π/4 + 2πk) * (3/π)x = (15/4 + 6k)So, the x-values where f(x) is maximum are x = 3/4 + 6k, and where f(x) is minimum are x = 15/4 + 6k, where k is any integer.Now, to find the minimum distance between an x₁ and x₂, we can take the smallest possible difference between these two sets of x-values. Let's look at the principal values when k=0.For maximum: x = 3/4For minimum: x = 15/4So, the distance between these two points is |15/4 - 3/4| = |12/4| = 3.Wait, but is this the minimum? Let me check with k=1 for x₂.If k=1, then x = 15/4 + 6 = 15/4 + 24/4 = 39/4. The distance from x=3/4 would be |39/4 - 3/4| = 36/4 = 9, which is larger.Similarly, for k=-1, x = 15/4 - 6 = 15/4 - 24/4 = -9/4. The distance from x=3/4 is | -9/4 - 3/4| = |-12/4| = 3, same as before.So, the minimum distance is 3. Therefore, the answer should be B: 3.But wait, let me think again. The function has a period, right? The period of sin(Bx + C) is 2π / |B|. Here, B is π/3, so the period is 2π / (π/3) = 6. So, the function repeats every 6 units.Therefore, the maximum and minimum values occur every half-period apart. So, half of 6 is 3. So, the distance between a maximum and the next minimum is 3. That makes sense.Alternatively, if I think about the graph of the sine function, it goes from maximum to minimum in half a period. So, the horizontal distance between a maximum and the nearest minimum is half the period, which is 3.So, yeah, I think 3 is the correct answer. So, the minimum value of |x₁ - x₂| is 3, which is option B.**Final Answer**boxed{B}

💡Okay, so I have this problem here: "If the graph of the power function ( f(x) ) passes through the point ( (2, frac{sqrt{2}}{2}) ), then ( f(9) = ______ )." Alright, let me try to figure this out step by step.First, I need to remember what a power function is. From what I recall, a power function is generally of the form ( f(x) = kx^n ), where ( k ) and ( n ) are constants. So, it's a function where the variable ( x ) is raised to some exponent, and there's a coefficient ( k ) in front. In this problem, they mention that the graph passes through the point ( (2, frac{sqrt{2}}{2}) ). That means when ( x = 2 ), ( f(x) = frac{sqrt{2}}{2} ). So, I can plug these values into the power function equation to find the constants ( k ) and ( n ).But wait, in the problem statement, they just say "the power function ( f(x) )", without specifying whether it has a coefficient or not. Hmm. I think sometimes power functions are considered to have the form ( f(x) = x^n ), without the coefficient ( k ). Maybe that's the case here? Let me check.If I assume that ( f(x) = x^n ), then plugging in the point ( (2, frac{sqrt{2}}{2}) ) gives me:( frac{sqrt{2}}{2} = 2^n )So, I need to solve for ( n ). Let me write that equation again:( 2^n = frac{sqrt{2}}{2} )Hmm, okay. I know that ( sqrt{2} ) is the same as ( 2^{1/2} ), and ( frac{1}{2} ) is ( 2^{-1} ). So, I can rewrite the right side as:( frac{sqrt{2}}{2} = frac{2^{1/2}}{2^1} = 2^{1/2 - 1} = 2^{-1/2} )So, now the equation is:( 2^n = 2^{-1/2} )Since the bases are the same, the exponents must be equal. Therefore:( n = -frac{1}{2} )Alright, so the power function is ( f(x) = x^{-1/2} ). Alternatively, that can be written as ( f(x) = frac{1}{x^{1/2}} ) or ( f(x) = frac{1}{sqrt{x}} ). All of these are equivalent.Now, the question asks for ( f(9) ). So, I need to plug ( x = 9 ) into the function:( f(9) = 9^{-1/2} )Again, let me think about how to compute this. ( 9^{-1/2} ) is the same as ( frac{1}{9^{1/2}} ). And ( 9^{1/2} ) is the square root of 9, which is 3. So, this simplifies to:( f(9) = frac{1}{3} )Wait, let me double-check my steps to make sure I didn't make a mistake. 1. I assumed the power function is ( f(x) = x^n ). Is that correct? Well, sometimes power functions can have a coefficient, but since the problem didn't specify, and it just said "the power function", I think it's safe to assume it's of the form ( x^n ). Plus, if there were a coefficient, we would need another point to determine both ( k ) and ( n ), but we only have one point here.2. Plugging in the point ( (2, frac{sqrt{2}}{2}) ) into ( f(x) = x^n ) gives ( frac{sqrt{2}}{2} = 2^n ). That seems right.3. Converting ( frac{sqrt{2}}{2} ) to exponents of 2: ( sqrt{2} = 2^{1/2} ) and ( frac{1}{2} = 2^{-1} ), so ( frac{sqrt{2}}{2} = 2^{1/2 - 1} = 2^{-1/2} ). That makes sense.4. Therefore, ( n = -1/2 ). So, the function is ( f(x) = x^{-1/2} ). That seems correct.5. Plugging in 9: ( f(9) = 9^{-1/2} = frac{1}{sqrt{9}} = frac{1}{3} ). Yep, that's right.I don't see any mistakes in my reasoning. So, I think the answer is ( frac{1}{3} ).But just to be thorough, let me consider if there's another way to approach this problem. Maybe by logarithms? Let's see.Starting again with ( 2^n = frac{sqrt{2}}{2} ). Taking the natural logarithm of both sides:( ln(2^n) = lnleft(frac{sqrt{2}}{2}right) )Using the logarithm power rule, ( ln(a^b) = bln(a) ), so:( n ln(2) = lnleft(frac{sqrt{2}}{2}right) )Simplify the right side:( lnleft(frac{sqrt{2}}{2}right) = ln(sqrt{2}) - ln(2) = frac{1}{2}ln(2) - ln(2) = -frac{1}{2}ln(2) )So, we have:( n ln(2) = -frac{1}{2}ln(2) )Divide both sides by ( ln(2) ):( n = -frac{1}{2} )Same result as before. So, that confirms that ( n = -1/2 ). Therefore, ( f(x) = x^{-1/2} ), and ( f(9) = 1/3 ).Another way to think about it is using exponent rules. Let me see:We have ( 2^n = frac{sqrt{2}}{2} ). Let me write both sides with exponents of 2:Left side: ( 2^n )Right side: ( frac{2^{1/2}}{2^1} = 2^{1/2 - 1} = 2^{-1/2} )So, ( 2^n = 2^{-1/2} implies n = -1/2 ). Yep, same answer.I think I've approached this from a few different angles now, and each time I arrive at the same conclusion. So, I feel confident that ( f(9) = frac{1}{3} ).Just to recap:1. Recognize that a power function is of the form ( f(x) = x^n ).2. Use the given point ( (2, frac{sqrt{2}}{2}) ) to set up the equation ( 2^n = frac{sqrt{2}}{2} ).3. Simplify the right side to ( 2^{-1/2} ).4. Since the bases are equal, set the exponents equal: ( n = -1/2 ).5. Write the function as ( f(x) = x^{-1/2} ).6. Substitute ( x = 9 ) into the function: ( f(9) = 9^{-1/2} = frac{1}{sqrt{9}} = frac{1}{3} ).All steps seem logical and consistent. I don't see any errors in my calculations or reasoning. So, I think I'm all set.

💡Okay, so I need to figure out what cos 120 degrees is equal to. Hmm, I remember that cosine is one of the basic trigonometric functions, and it relates to the adjacent side over the hypotenuse in a right triangle. But 120 degrees is more than 90 degrees, so it's in the second quadrant. I think in the second quadrant, cosine values are negative because cosine corresponds to the x-coordinate on the unit circle, and in the second quadrant, x is negative.Alright, so cos 120 degrees should be negative. Now, I need to find the exact value. I remember that 120 degrees is related to 60 degrees because 120 is 180 minus 60. Maybe I can use some kind of identity or reference angle here. The reference angle for 120 degrees would be 180 minus 120, which is 60 degrees. So, cos 120 degrees should be related to cos 60 degrees, but since it's in the second quadrant, it should be negative.I recall that cos 60 degrees is 0.5 or 1/2. So, if cos 120 degrees is related to cos 60 degrees but negative, then cos 120 degrees should be -1/2. Let me double-check that. If I think about the unit circle, at 60 degrees, the coordinates are (0.5, sqrt(3)/2). At 120 degrees, which is in the second quadrant, the x-coordinate should be negative, so it would be (-0.5, sqrt(3)/2). Therefore, the cosine of 120 degrees is indeed -0.5, which is -1/2.Wait, but let me make sure I didn't mix up sine and cosine. Sine corresponds to the y-coordinate, right? So at 60 degrees, sine is sqrt(3)/2, and at 120 degrees, sine is still positive because y is positive in the second quadrant. But cosine is negative in the second quadrant, so yes, cos 120 degrees is -1/2.I think that's correct. So, looking at the options, A is -1/2, which matches what I found. B is -sqrt(3)/2, which is the sine of 120 degrees, not cosine. C and D are positive, so they can't be right because cosine is negative in the second quadrant. So, the answer should be A: -1/2.

💡To find the greatest two-digit multiple of 17, I will start by identifying how many times 17 fits into the largest two-digit number, which is 99.First, I divide 99 by 17 to determine the maximum integer multiplier:[frac{99}{17} approx 5.8235]Since we need an integer multiplier, I take the floor of 5.8235, which is 5.Next, I multiply 17 by this integer to find the greatest two-digit multiple:[17 times 5 = 85]Finally, I verify that 85 is indeed a two-digit number and a multiple of 17, confirming that it is the greatest two-digit multiple of 17.

💡Okay, so I have this problem here about 150 students at a workshop, and each student can write poetry, paint, or play an instrument. Some can do more than one, but none can do all three. The problem gives me some numbers: 80 can't write poetry, 90 can't paint, and 60 can't play an instrument. I need to find how many students have exactly two skills.Hmm, let me break this down. First, if there are 150 students total, and 80 can't write poetry, that means 150 - 80 = 70 students can write poetry. Similarly, 150 - 90 = 60 can paint, and 150 - 60 = 90 can play an instrument.So, we have 70 poets, 60 painters, and 90 musicians. Now, if I add those up, that's 70 + 60 + 90 = 220. Wait, but there are only 150 students. That means some students are counted more than once because they have multiple skills.Since no one has all three skills, the overlaps can only be between two skills. So, the total number of students with at least one skill is 150, but the sum of individual skills is 220. The difference, 220 - 150 = 70, must be the number of students who have exactly two skills because each of these students is counted twice in the total of 220.Let me make sure I'm not missing anything. Each student who has exactly two skills is counted twice in the total of 220, so subtracting the total number of students gives me the number of overlaps, which corresponds to the number of students with exactly two skills. Yeah, that makes sense.So, the answer should be 70 students who have exactly two skills.

💡Okay, so I have this question about the composition of air and what kind of statistical graph would be best to represent that information. Let me try to break this down step by step.First, the question mentions that air is composed of approximately 78% nitrogen, 21% oxygen, and about 1% other trace gases. It's asking which type of graph would be suitable to reflect this information. The options are a bar graph, a line graph, a pie chart, or a frequency distribution histogram.Alright, let's start by recalling what each of these graphs is typically used for.A bar graph is usually used to compare quantities across different categories. For example, if I wanted to compare the sales of different products over a month, a bar graph would be useful because it allows for easy visual comparison between the products.A line graph, on the other hand, is more suited for showing trends over time. If I wanted to see how the temperature changes throughout the day, a line graph would be appropriate because it can illustrate the continuous change in temperature.A pie chart is used to show the proportions of a whole. It's particularly useful when you want to display how different parts contribute to the whole. For instance, if I wanted to show the percentage distribution of different expenses in a budget, a pie chart would be ideal because it visually represents each category's proportion relative to the total.Lastly, a frequency distribution histogram is used to show the distribution of numerical data. It's similar to a bar graph but is specifically used for showing the frequency of data within certain ranges. For example, if I wanted to show how many people scored within certain grade ranges on a test, a histogram would be the way to go.Now, looking back at the question, we're dealing with the composition of air, which is given in percentages. The key here is that these percentages add up to 100%, representing the whole of the air composition. We're not comparing different categories over time or across different products; instead, we're showing how each gas contributes to the whole composition of air.Given that, a pie chart seems like the most appropriate choice because it's designed to show the proportions of a whole. Each slice of the pie would represent the percentage of each gas in the air, making it easy to visualize how much of the air is nitrogen, oxygen, and other trace gases.Just to double-check, let's consider the other options again. A bar graph could technically show the percentages, but it wouldn't convey the idea that these percentages make up the whole as effectively as a pie chart. A line graph isn't suitable here because we're not dealing with data over time or a continuous interval. And a frequency distribution histogram doesn't apply because we're not dealing with numerical data ranges or frequencies.Therefore, after considering all the options and the nature of the data, I'm confident that a pie chart is the best choice for representing the composition of air in this case.