Hello, Jacobpm64!
It's easier than you think . . .
The hypotenuse of a right triangle has one end at the origin
and one end on the curve , with
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. At what xvalue does it occur? Code:

 *
 *
 * 
 * y
 * 
*+
 x
The area of a triangle is:
The area of this triangle is: . . . . where
So we have: .
Differentiate: .
Equate to zero: .
Set each factor equal to zero and solve.
. . . . . This gives minimum area ... ha!
. . . . . but can never equal zero.
. . . . . There!
Maximum area occurs when and
The maximum area is: .