Question

The hypotenuse of a right triangle has one end at the origin and one end on the curve y=x^(7)e^(-7x) , with x greater than or equal to 0. One of the other two sides is on the x-axis, the other side is parallel to the y-axis.

Find the maximum area of such a triangle. Round your answer to 4 decimal places.
Maximum Area =

At what x-value does it occur?

Any help on these two questions would be greatly appreciated.

Answer 1

Area of triangle = 1/2(base)(height)
= 1/2(x)(x^7)(e^(-7x)
= 1/2(x^8)(e^(-7x)

Area' = 1/2[x^8(-7e^(-7x)) + 8x^7(e^(-7x))]
= 0 for a max/min of Area
(1/2)e^(-7x)x^7[-7x + 8] = 0
-7x + 8 = 0
x = 8/7 or x = 1.142857

so max area = .000488145

test: take a slighter larger and a slightly smaller value of x

if x = 1.14
area = .000488133

if x = 1.15
area = .00048807

so the largest area correct to 4 decimals is .0005 and it occurs when x = 8/7