Consider a sphere with given radius . Find the values for the radius and altitude ( and , respectively) of the right circular cylinder of maximum lateral surface area (given by ) that can be inscribed in the sphere.

First, we want to write as a function of . From the diagram, we see that we have a right triangle with hypotenuse of length and legs of lengths and (since the second leg only goes to the center of the sphere, not the full length of the cylinder). Therefore,

Then, we are given that the lateral surface area, , is given by the formula

Calling this function and differentiating we find,

Setting this equal to 0 we have,

Since

this point is a minimum. Hence, the lateral surface area is minimal when

We need to find the cylinder with the MAXIMUM lateral surface. Your derivative sign is incorrect for the intervals to the left and to the right of the critical point.