Introduction to Optimization Problems

A quantity like area, volume, or cost changes as some length or rate changes. At the best possible value, the graph of that quantity is at a peak or a valley, and at a smooth peak or valley the tangent line is flat. That is why the maximum or minimum almost always happens where the derivative equals zero, or occasionally at an endpoint of the allowed interval. Optimization problems just ask you to find that flat spot.

The real work is setup. Read the problem and name the quantity you want to optimize, usually area, volume, distance, or cost. Write a formula for it, but it will usually involve two variables, because real shapes have two dimensions. Find a constraint sentence hiding in the problem, like a fixed perimeter or a fixed amount of fencing, and use it to solve for one variable in terms of the other. Substitute that into your formula so you get one function of one variable. Then take the derivative, set it equal to zero, solve, and check that this gives a maximum, not a minimum, using the second derivative test or by checking sign changes.

This topic is where every tool from earlier in the unit comes together: finding critical points, using the first or second derivative test, and checking endpoints on a closed interval. Everything after this is the same process applied to messier shapes, like cylinders, cones, or distances between moving points, so getting comfortable with the four-step setup here, name the quantity, write the constraint, substitute, differentiate, pays off for the rest of the unit.