Help with calculus optimization problem?

[lovelylilyberry]

I need to write my own optimization problem, and here's what I have so far:

A contractor wants to build an aboveground, cylindrical swimming pool. The client wants the pool to hold 148 cubed feet of water. If he wants the swimming pool to use the least amount of material as possible, what should be the pool’s height?

The variables in this equation are the radius, r, and the height, h.

The equation for the volume of the pool is V = πr²h = 148. To make it so we are only working with one variable, solve this equation for height.

V = πr²h = 148
h = 148/πr²

The expression that needs to be minimized is for the surface area of the pool:

SA = πr² + 2πrh.

I've already figured out the critical number for r and confirmed it's a minimum.
But my question is how do I find the endpoints of this problem, so I can test them too? I figure one endpoint is r = 0, but how do I find the other?

 

[MarkFL]

You really have no lower or upper bound for r, since as h goes to zero, r grows without bound and likewise as r goes to zero, h grows without bound as the two share an inverse relationship. I would simply be content to demonstrate that for all positive and real r, the second derivative is positive, so you know your minimum is a global minimum.