💡Okay, so I have this problem where I'm given a function f(x) = 8 + 2x - x², and another function g(x) which is defined as f(2 - x²). I need to figure out whether g(x) is increasing or decreasing on certain intervals given in the options A to D.First, I think I should find out what g(x) actually is. Since g(x) is f evaluated at (2 - x²), I can substitute (2 - x²) into f(x). So, let me write that down:g(x) = f(2 - x²) = 8 + 2*(2 - x²) - (2 - x²)².Now, I need to simplify this expression step by step. Let me compute each part:First, compute 2*(2 - x²):2*(2 - x²) = 4 - 2x².Next, compute (2 - x²)²:(2 - x²)² = 4 - 4x² + x⁴.So, putting it all together:g(x) = 8 + (4 - 2x²) - (4 - 4x² + x⁴).Now, let me simplify this:g(x) = 8 + 4 - 2x² - 4 + 4x² - x⁴.Combine like terms:8 + 4 - 4 = 8,-2x² + 4x² = 2x²,and then we have -x⁴.So, g(x) simplifies to:g(x) = -x⁴ + 2x² + 8.Okay, so now I have g(x) expressed as a quartic function: -x⁴ + 2x² + 8.To determine whether g(x) is increasing or decreasing on certain intervals, I need to find its derivative, g'(x). The derivative will tell me the slope of the function at any point x, which indicates whether the function is increasing (positive slope) or decreasing (negative slope).So, let's compute the derivative of g(x):g'(x) = d/dx [-x⁴ + 2x² + 8].The derivative of -x⁴ is -4x³,the derivative of 2x² is 4x,and the derivative of 8 is 0.So, putting it together:g'(x) = -4x³ + 4x.I can factor this expression to make it easier to analyze:g'(x) = -4x(x² - 1).Wait, let me check that factoring:-4x³ + 4x = -4x(x² - 1). Yes, that's correct.Now, let's factor (x² - 1) further since it's a difference of squares:x² - 1 = (x - 1)(x + 1).So, the derivative becomes:g'(x) = -4x(x - 1)(x + 1).This is helpful because it shows the critical points where the derivative is zero, which are x = -1, x = 0, and x = 1. These points divide the real number line into intervals where the sign of the derivative (and hence the increasing/decreasing nature of g(x)) can be determined.The intervals are:1. (-∞, -1)2. (-1, 0)3. (0, 1)4. (1, ∞)To determine the sign of g'(x) in each interval, I can pick a test point in each interval and plug it into the derivative expression.Let's go through each interval:1. Interval (-∞, -1): Let's choose x = -2. Plugging into g'(x): g'(-2) = -4*(-2)*(-2 - 1)*(-2 + 1) = -4*(-2)*(-3)*(-1) = -4*(-2)*3 = -4*(-6) = 24, which is positive. So, g(x) is increasing on (-∞, -1).2. Interval (-1, 0): Let's choose x = -0.5. Plugging into g'(x): g'(-0.5) = -4*(-0.5)*(-0.5 - 1)*(-0.5 + 1) = -4*(-0.5)*(-1.5)*(0.5) = -4*(-0.5)*(-1.5)*0.5 Let's compute step by step: First, -4 * (-0.5) = 2 Then, 2 * (-1.5) = -3 Then, -3 * 0.5 = -1.5 So, g'(-0.5) = -1.5, which is negative. Therefore, g(x) is decreasing on (-1, 0).3. Interval (0, 1): Let's choose x = 0.5. Plugging into g'(x): g'(0.5) = -4*(0.5)*(0.5 - 1)*(0.5 + 1) = -4*(0.5)*(-0.5)*(1.5) Let's compute step by step: First, -4 * 0.5 = -2 Then, -2 * (-0.5) = 1 Then, 1 * 1.5 = 1.5 So, g'(0.5) = 1.5, which is positive. Wait, that's positive. But wait, let me double-check my calculations because I might have made a mistake. Wait, let's recompute: g'(0.5) = -4*(0.5)*(0.5 - 1)*(0.5 + 1) = -4*(0.5)*(-0.5)*(1.5) Compute each part: -4 * 0.5 = -2 -2 * (-0.5) = 1 1 * 1.5 = 1.5 So, yes, it's positive. Hmm, but according to the initial analysis, I thought it was negative. Maybe I made a mistake in the sign. Wait, let me check the derivative again: g'(x) = -4x(x - 1)(x + 1) At x = 0.5: -4*(0.5) = -2 (0.5 - 1) = -0.5 (0.5 + 1) = 1.5 So, multiplying all together: -2 * (-0.5) * 1.5 -2 * (-0.5) = 1 1 * 1.5 = 1.5, which is positive. Hmm, so according to this, g'(0.5) is positive, meaning g(x) is increasing on (0,1). But I thought earlier it was decreasing. Maybe I made a mistake in the initial analysis. Wait, let me think again. The derivative is positive, so the function is increasing. So, perhaps my initial thought was wrong. Wait, but let me check another point in (0,1). Let's take x = 0.25. g'(0.25) = -4*(0.25)*(0.25 - 1)*(0.25 + 1) = -4*(0.25)*(-0.75)*(1.25) Compute step by step: -4 * 0.25 = -1 -1 * (-0.75) = 0.75 0.75 * 1.25 = 0.9375, which is positive. So, yes, it's positive. Therefore, g(x) is increasing on (0,1). But wait, that contradicts my initial thought. Maybe I made a mistake in the initial analysis. Wait, perhaps I made a mistake in the sign when factoring. Let me re-examine the derivative: g'(x) = -4x³ + 4x = -4x(x² - 1) = -4x(x - 1)(x + 1) So, the critical points are at x = -1, 0, 1. Now, let's analyze the sign of each factor in each interval: For x in (-∞, -1): x is negative, (x - 1) is negative, (x + 1) is negative. So, -4x is positive (since x is negative), (x - 1) is negative, (x + 1) is negative. So, overall: positive * negative * negative = positive. For x in (-1, 0): x is negative, (x - 1) is negative, (x + 1) is positive. So, -4x is positive, (x - 1) is negative, (x + 1) is positive. So, overall: positive * negative * positive = negative. For x in (0, 1): x is positive, (x - 1) is negative, (x + 1) is positive. So, -4x is negative, (x - 1) is negative, (x + 1) is positive. So, overall: negative * negative * positive = positive. For x in (1, ∞): x is positive, (x - 1) is positive, (x + 1) is positive. So, -4x is negative, (x - 1) is positive, (x + 1) is positive. So, overall: negative * positive * positive = negative. Wait, so according to this, in (0,1), the derivative is positive, so g(x) is increasing. But in the initial analysis, I thought it was decreasing. So, perhaps I made a mistake earlier. Let me check again with x = 0.5: g'(0.5) = -4*(0.5)*(0.5 - 1)*(0.5 + 1) = -4*(0.5)*(-0.5)*(1.5) = (-2)*(-0.5)*(1.5) = (1)*(1.5) = 1.5, which is positive. So, yes, it's positive. Therefore, g(x) is increasing on (0,1). So, my initial thought that it was decreasing was wrong. Wait, but in the problem statement, option B says "is a decreasing function on the interval (0,1)", which would be incorrect based on this analysis. So, perhaps I need to re-examine. Wait, maybe I made a mistake in the derivative. Let me double-check the derivative: g(x) = -x⁴ + 2x² + 8 So, g'(x) = -4x³ + 4x. Yes, that's correct. So, the derivative is positive in (0,1), meaning g(x) is increasing there. So, option B is incorrect. Similarly, for interval (1, ∞), the derivative is negative, so g(x) is decreasing there. So, summarizing: - (-∞, -1): increasing - (-1, 0): decreasing - (0, 1): increasing - (1, ∞): decreasing Now, looking back at the options: A: decreasing on (-1, 0) – correct B: decreasing on (0,1) – incorrect, it's increasing C: increasing on (-2, 0) – Well, (-2,0) includes (-2, -1) and (-1,0). From our analysis, on (-∞, -1), it's increasing, and on (-1,0), it's decreasing. So, on (-2,0), the function is increasing on (-2, -1) and decreasing on (-1,0). So, overall, it's not entirely increasing on (-2,0). So, C is incorrect. D: increasing on (0,2) – Well, (0,2) includes (0,1) and (1,2). On (0,1), it's increasing, but on (1,2), it's decreasing. So, overall, it's not entirely increasing on (0,2). So, D is incorrect. Therefore, the correct answer is A. Wait, but earlier I thought that in (0,1), it's increasing, so option B is wrong, and option D is also wrong because after x=1, it starts decreasing. So, the only correct option is A: decreasing on (-1,0). Therefore, the answer is A.

💡Okay, so I have this problem about toothpicks arranged in a pattern. It starts with 5 toothpicks, and each stage after that adds 4 more toothpicks. I need to find out how many toothpicks are needed for the 100th stage. Hmm, let's break this down step by step.First, I know that the pattern starts with 5 toothpicks. That's the initial stage, which I can consider as stage 1. Then, each subsequent stage adds 4 toothpicks. So, stage 2 would have 5 + 4 = 9 toothpicks, stage 3 would have 9 + 4 = 13 toothpicks, and so on. It seems like each time, we're adding the same number of toothpicks, which is 4. That makes me think this is an arithmetic sequence because the difference between consecutive terms is constant.In arithmetic sequences, there's a formula to find the nth term. The formula is:[ a_n = a_1 + (n - 1) cdot d ]Where:- ( a_n ) is the nth term,- ( a_1 ) is the first term,- ( d ) is the common difference,- ( n ) is the term number.In this problem:- ( a_1 = 5 ) (since the first stage has 5 toothpicks),- ( d = 4 ) (because we add 4 toothpicks each time),- ( n = 100 ) (we need the number for the 100th stage).Plugging these values into the formula:[ a_{100} = 5 + (100 - 1) cdot 4 ]Let me compute ( 100 - 1 ) first, which is 99. Then multiply that by 4:[ 99 times 4 = 396 ]Now, add that to the initial 5 toothpicks:[ 5 + 396 = 401 ]So, according to this calculation, the 100th stage would require 401 toothpicks. Wait, let me double-check to make sure I didn't make a mistake. Starting from stage 1 with 5 toothpicks, each stage adds 4. So, stage 2 is 9, stage 3 is 13, and so on. The pattern increases by 4 each time, which confirms it's an arithmetic sequence. The formula I used is correct for finding the nth term in such a sequence. Another way to think about it is that for each stage after the first, we're adding 4 toothpicks. So, from stage 1 to stage 100, there are 99 intervals where we add 4 toothpicks each. Therefore, the total number of toothpicks added after the first stage is ( 99 times 4 = 396 ). Adding the initial 5 gives ( 5 + 396 = 401 ). Yeah, that seems right. I don't think I made any calculation errors here. So, I'm pretty confident that the 100th stage requires 401 toothpicks.

💡To determine how much more sandwich the first participant ate compared to the second one, I need to subtract the fraction of the sandwich eaten by the second participant from the fraction eaten by the first participant.First, I'll identify the fractions involved:- The first participant ate (frac{5}{6}) of a sandwich.- The second participant ate (frac{2}{3}) of a sandwich.Next, I need to subtract these two fractions. To do this, I should ensure both fractions have the same denominator. The denominators here are 6 and 3. The least common multiple of 6 and 3 is 6, so I'll convert (frac{2}{3}) to a fraction with a denominator of 6.[frac{2}{3} = frac{2 times 2}{3 times 2} = frac{4}{6}]Now, both fractions have the same denominator:[frac{5}{6} - frac{4}{6} = frac{5 - 4}{6} = frac{1}{6}]Therefore, the first participant ate (frac{1}{6}) more of the sandwich than the second participant.

💡First, I need to find the greatest multiple of 3 that is less than 1000 and also a multiple of 5 and 7. To do this, I'll start by determining the least common multiple (LCM) of the numbers 3, 5, and 7.Since 3, 5, and 7 are all prime numbers, their LCM is simply the product of these numbers:[ text{LCM}(3, 5, 7) = 3 times 5 times 7 = 105 ]Next, I need to find the largest multiple of 105 that is less than 1000. I'll divide 1000 by 105 to find how many times 105 fits into 1000:[ 1000 div 105 approx 9.523 ]Since I can't have a fraction of a multiple, I'll take the integer part of the division, which is 9. This means the largest multiple of 105 under 1000 is:[ 105 times 9 = 945 ]Therefore, the greatest multiple of 3 that is less than 1000 and also a multiple of 5 and 7 is 945.

💡To find the points where circle ( C ) intersects the ( y )-axis, we start by setting ( x = 0 ) in the equation of the circle. The equation of circle ( C ) with center ((-4, 5)) and radius ( 9 ) units is:[(x + 4)^2 + (y - 5)^2 = 81]Substituting ( x = 0 ) into the equation:[(0 + 4)^2 + (y - 5)^2 = 81 16 + (y - 5)^2 = 81 (y - 5)^2 = 65 y - 5 = pm sqrt{65} y = 5 pm sqrt{65}]Thus, the ( y )-coordinates of the intersection points are ( 5 + sqrt{65} ) and ( 5 - sqrt{65} ). Adding these ( y )-coordinates:[(5 + sqrt{65}) + (5 - sqrt{65}) = 10]Therefore, the sum of the ( y )-coordinates of the intersection points is ( boxed{10} ).

💡First, I will examine the figure to identify the different rectangular sections that make up the entire shape.Next, I will determine the dimensions of each rectangle by using the given unit lengths. This involves identifying the height and width of each section based on the numbers provided.Once I have the dimensions of each rectangle, I will calculate the area of each one by multiplying its height by its width.Finally, I will sum the areas of all the individual rectangles to find the total area of the entire figure.