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 $f$ ...