# math

Show that if a rectangle has its base on the x-axis and two of its vertices on the curve y = e^-x^2 , then the rectangle will have the largest possible area when the two vertices are at the points of inflection of the curve.

1. 👍
2. 👎
3. 👁
4. ℹ️
5. 🚩
1. let (x,y) be the point of contact in quadrant I, then (-x,y) is the other vertex.
base of rectange = 2x
height of rectange = y = e^(-x^2)

area = 2xy
= 2x(e^(-x^2))
d(area)/dx = 2x(e^(-x^2))(-2x) + 2e(-x^2)
= 0 for a max of area
2e^(-x^2)[-2x^2 + 1] = 0
2e^(-x^2) = 0 ---> no solution or

-2x^2 + 1 = 0
x = ± 1/√2
then y = e^(-1/2) = 1/√e
max area occurs when vertices are (1/√2, 1/√e) and (-1/√2, 1/√e)

I will leave it up to you to differentiate
y = e^(-x^2) twice, set that equal to zero, and show that x = ±1/√2

1. 👍
2. 👎
3. ℹ️
4. 🚩

## Similar Questions

1. ### math

Note: Enter your answer and show all the steps that you use to solve this problem in the space provided. An image of a rocket is shown. The rocket is made up of a triangle, a rectangle, and a trapezoid. The triangle at the top of

2. ### Calculus AP

Let R be the region in the first quadrant bounded by the graph y=3-√x the horizontal line y=1, and the y-axis as shown in the figure to the right. Please show all work. 1. Find the area of R 2. Write but do not evaluate, an

3. ### geometry

The vertices of a triangle are P(-8,1), Q(-6,-8), and R(4,-3) Name the vertices of the image reflected across the x-axis. my answer; P1(-8,1), Q1(-6,8), and R1(4,3)

4. ### Geometry

The vertices of a triangle are P(-4,1), Q(-2,-8), R(8,-1). What are the vertices of the image reflected across the y-axis? my answer is P'(4,1),Q'(2,-8),R'(-8,-1)

1. ### Math

What is the area of the largest rectangle with lower base on the x-axis and upper vertices on the curve y equals= 12 minus x^2?

2. ### Math

1. The vertices of a triangle are P(2,-4), Q(-5,3), and R(-1,-2). What are the vertices of the image reflected across the y-axis? A) P(2,-4), Q(-5,3), R(-1,-2) B) P(2,4), Q(-5,-3), R(-1,2) C) P(-2,-4), Q(5,3), R(1,-2) D) P(-2,4),

3. ### calculus

A rectangle has its base on the x-axis and its 2 upper corners on the parabola y=12-x^2. What is the largest possible area of the rectangle?

4. ### Calculus

A rectangle has a diagonal of 8cm. The diagonal creates a 60 degrees angle at the base of the rectangle. Write an exact expression for the base and the height of the rectangle. ^^ Don't get how to do that, Ive only drawn a

1. ### Calculus :)

A rectangle is inscribed with its base on the x-axis and its upper corners on the parabola y=6–x^2. What are the dimensions of such a rectangle with the greatest possible area? Find Width=____ & Height=4 just need to find

2. ### Math

geometry- On the coordinate plane, draw a rectangle ABCD with vertices at A (1,4), B(5,4), C(5,1) and D (1,1). Then graph and describe the new rectangle formed when you subtact 3 from each coordinate of the vertices in rectangle

3. ### calculus

An isosceles triangle is drawn with its vertex at the origin and its base parallel to the x-axis. The vertices of the base are on the curve 5y=25-x^2 Find the area of the largest such triangle.

4. ### calculus

a rectangle has its base on the x-axis, and its upper corners in the graph of y=27-x^2. what is the maximal area of this rectangle?