Question

Find the area of the largest rectangle that can be drawn with one side along the x-axis and two vertices on the curve with equation y = 1/(x^2 + 1)

Answer 1

If you replace x with -x you get the same y value and therefore the graph is symmetric about the y axis.

Let (x,y) be a point on the curve.
The length of the rectangle formed will be 2x measured along the x-axis and the height will be y measured along the y axis.

Area A = 2xy

Put y = 1/(x^2 + 1) in A

To maximize A, find dA/dx, set it to 0 and solve for x.
Put that x value and find y from y =1/(x^2 + 1)

Put the x,y values in A = 2xy and compute the maximum area A.

Answer 2

ok i did that and i got x = 1 and x = -1 and i neglected the -1. but when i test both sides for when x = 1 i see that it is decreasing on both sides which means that there is no local min or local max. and i am more interested in finding the local max. could i just be doing an arithmetic error when i test x=1 from both sides?

Answer 3

The problem here is not to find the maximum or the minimum for the curve.
The question here is to maximize the area formed by the rectangle.
So no point finding dy/dx. They need dA/dx = 0.

Answer 4

ohhh ok. Thank you for clearing things up

Answer 5

Your x = 1 is correct. So y = 1/2. Max area = 1.