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I have been struggling with this one for a wile:

A right circular cylinder is inscribed in a sphere of radius R. Find the largest surface area of a cylinder.

I could do the one where I have to obtain the maximum volume, since it is possible to chose the variables to get a "clean" derivative equation, but with a square root which is a little more tricky.

S=2.Pi.r.h + 2.Pi.r^2

Set:[y=r=(R^2-x^2)^1/2; h=2.x]

S= 4.Pi.(R^2-x^2)^1/2 .x + 2.Pi.(R^2-x^2)

dS/Dx=2.Pi[2(R^2-x^2)^1/2 - 2x^2.(R^2-x^2)^-1/2 - 2x]

How do I go to work to solve this one, as the sqrt. is a pesky bugger.

Setting this to 0 is the same as -x/sqrt(R²-x²) + sqrt(R²-x²)) + 2x = 0.

Multiply through by sqrt(R²-x²), giving -x + R²-x² + 2x*sqrt(R²-x²) = 0.

The is the same as R² - x² - x = -2x*sqrt(R²-x²).

Dividing by -2x gives -R²/(2x) + x/2 + 1 = -sqrt(R²-x²).

To solve, square both sides and use numerical analisys to find x.

Sorry I don't have time, but I'm currently trying to fix an error on the PC.

Calculus

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