prove that out of the rectangles under the curve = e raised to the power of -xsquared on the x axis and upper vertices on the curve, the one has the largest area the two upper vertices of which are exactly at the points of inflection of the curve.
To prove that the rectangle with its upper vertices exactly at the points of inflection of the curve y = e^(-x^2) has the largest area among all rectangles under the curve, we can follow these steps:
1. Find the equation of the curve: We are given that the curve is y = e^(-x^2).
2. Determine the points of inflection: Points of inflection occur where the concavity of the curve changes. To find these points, we need to find the second derivative of the curve and solve for x when the second derivative equals zero.
- First derivative: y' = -2xe^(-x^2)
- Second derivative: y'' = (-2 + 4x^2)e^(-x^2)
- Setting y'' = 0, we have: (-2 + 4x^2)e^(-x^2) = 0
- Solving the equation, we find two points of inflection: x = -1 and x = 1.
3. Determine the y-values at the points of inflection: Plug the x-values (-1 and 1) into the equation of the curve to find the corresponding y-values.
- For x = -1: y = e^(-(-1)^2) = e^(-1) ≈ 0.368
- For x = 1: y = e^(-(1)^2) = e^(-1) ≈ 0.368
4. Construct the rectangle: The rectangle we are interested in has its upper vertices at the points of inflection of the curve. Therefore, the height of the rectangle is given by the y-values at the points of inflection, which are approximately 0.368.
5. Determine the width of the rectangle: The width of the rectangle is the distance between the x-values of the points of inflection, which is 1 - (-1) = 2.
6. Calculate the area of the rectangle: The area of a rectangle is given by the product of its width and height.
- Area = Width * Height = 2 * 0.368 ≈ 0.736
7. Compare with other rectangles: To demonstrate that this rectangle has the largest area, we need to compare it with other rectangles under the curve with different widths.
- Let's consider a rectangle with a width of 4 units (x = -2 to x = 2). Its maximum height would occur at the center of the rectangle, where x = 0.
- Height = y(0) = e^(-(0)^2) = e^0 = 1
- Area = Width * Height = 4 * 1 = 4
8. Conclusion: Comparing the areas of the two rectangles, we find that the rectangle with the upper vertices at the points of inflection has a larger area (0.736) compared to the rectangle with the maximum height at the center (4). Therefore, we can conclude that the rectangle with its upper vertices at the points of inflection has the largest area among all rectangles under the curve y = e^(-x^2) on the x-axis.