Derivatives

In this lesson, we’re going to learn how to use derivatives to solve optimization problems. The derivative of a function tells us a lot of information about that function, including what points could be the local minima or maxima.

Local minima and maxima

Consider the following function:

Points in the colors red and green look quite special. See what happens with the tangent line at those points. The tangent is flat. When a line is flat, its slope is zero.

The slope being zero means the same as $f'(x) = 0$. It corresponds to the points when $f$ reaches a local minimum or maximum. Those are the interesting points to us! A local minimum is a point such that there’s no other point around it where $f$ has a lower value. Analogously, a local maximum is a point such that no point around it makes $f$ greater. The minimum among all local minima of $f$ is the global minimum of $f$ and the maximum among all local maxima of $f$ is the global maximum.

All points where $f'(x) = 0$ are candidates to be a local minimum or a local maximum. So the first thing to do is to find all those points. But how can we differentiate a minimum from a maximum? Well, let’s look again at the last figure.

At a local minimum, the function stops going down and begins going up. In a local maximum, the function does quite the opposite. Then, to differentiate a minimum from a maximum, we just need to analyze the sign of $f'$ before and after the point.

For example, consider $f(x) = x^2$ ...