13.1 Minima, maxima, and derivatives
Intuitively, the notion of minima and maxima is simple. Take a look at Figure 13.1 below.
Peaks of hills are the maxima, and the bottoms of valleys are the minima. Minima and maxima are collectively called extremal or optimal points. As our example demonstrates, we have to distinguish between local and global optima. The graph has two valleys, and although both have a bottom, one of them is lower than the other.
The really interesting part is finding these, as we’ll see next. Let’s consider our example above to demonstrate how derivatives are connected to local minima and maxima.
If we use our geometric intuition, we see that the tangents are horizontal at the peaks of the hills and the bottoms of the valleys. Intuitively, if the tangent line is not horizontal, then there’s an elevation or decline in the graph. This is illustrated by Figure 13.2.
