A box with a square base and open top must have a volume of 202612 $\displaystyle {c}{m}^{{3}}$. We wish to find the dimensions of the box that minimize the amount of material used.

First, find a formula for the surface area of the box in terms of only $\displaystyle {x}$, the length of one side of the square base.


$\displaystyle {A}{\left({x}\right)}=$Preview Question 6 Part 1 of 6  

Next, find the derivative, $\displaystyle {A}'{\left({x}\right)}$.
$\displaystyle {A}'{\left({x}\right)}=$Preview Question 6 Part 2 of 6  

The critical value is $\displaystyle {x}=$

The function is Select an answer decreasing increasing until the critical value, and Select an answer decreasing increasing after, so the critical value corresponds to a local Select an answer maximum minimum .