Find the radius of a cylinder of given volume V if its surface area is a minimum. this question is driving me crazy as I'm not sure how they've got the answer.
The surface area is given as $S = 2\pi r^2 + \frac {1}{50r} $ and they are asking for the value of r for which S is minimum.
The derivative of this (I hope!) is $4\pi r - \frac {1}{50r^2}$
Then to find the value of r when S is a minimum I presume you set the derivative to equal $0$.
The book shows the value $200π - \frac 13 $ but I'm not sure how they've got this figure from setting the derivative to $0$.
Any insight would be appreciated!
Edit: My bad, answer is raised to negative $ \frac {1}{3}$, you live, you learn. Thanks for pointing this out.

By the classical formulas, $$V=\pi r^2h$$ and $$S=2\pi r^2+2\pi rh.$$
Eliminating $h$,
$$S=2\pi r^2+\frac{2V}r.$$
Then cancelling the derivative,
$$4\pi r-\frac{2V}{r^2}=0$$ or $$r^3=\frac V{2\pi}.$$
