Maxima and minima of differentiable functions are always attained at either boundary points (the largest or smallest value of possible, for instance) or points at which the tangent line to the curve is horizontal, i.e., .
(1) To find the surface area of a tank in terms of its length , we begin by writing down what we know:
Since , we have
All that remains is to substitute these expressions in the first equation. To find the value of that allows us minimum surface area, we look for points at which :
As we know that is always positive and that the surface area would tend to as approached or , we can be sure that the mininum is not attained at a boundary point. All we must do, then, is find the solution of .
(3) Here's a hint:
To find in terms of , we may begin by finding in terms of .
Hope this helps!