Optimization problems are one application of the derivative. They often focus on geometric shapes but any required formulas, such as equations for area or volume, will be provided. After choosing appropriate equations and defining variables (sometimes one in terms of another) you take the derivative of the equation for the value(s) in question. The point at which the derivative of the equation is equal to zero will give the extreme value you are looking for. With this general procedure in mind let's try some problems.
The first two problems have step-by-step solutions; the last two have only answers.
Problem 1 (Given dimensions, find volume)

Optimization problems are one application of the derivative. They often focus on geometric shapes but any required formulas, such as equations for area or volume, will be provided. After choosing appropriate equations and defining variables (sometimes one in terms of another) you take the derivative of the equation for the value(s) in question. The point at which the derivative of the equation is equal to zero will give the extreme value you are looking for. With this general procedure in mind let's try some problems.

The first two problems have step-by-step solutions; the last two have only answers.

Problem 1 (Given dimensions, find volume)

Problem 2 (Given volume, find dimensions)

Two more problems (with no solutions, just answers)

Resources

Paul's Notes on Optimization

Many Problems with solutions

Process with examples

Multistep Process

Page with Interactive tutorial