Calculus
posted by George .
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 xaxis, the other side is parallel to the yaxis.
Find the maximum area of such a triangle. Round your answer to 4 decimal places.
Maximum Area =
At what xvalue does it occur?
Any help on these two questions would be greatly appreciated.

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
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