4.1 Section Summary

In this section we examined what we can learn about the graph of a function from a formula for the derivative. In particular, if the graph of `f` has a point $(x_0,y_0)$ that is higher than any nearby points, then either $x_0$ is an end point of the domain or $f^{\prime }\left(x_0\right)=0.$ Similarly, if the graph of `f` has a point $(x_0,y_0)$ that is lower than any nearby points, then either $x_0$ is an end point of the domain or $f^{\prime }\left(x_0\right)=0.$ Thus the natural candidates for maximum and minimum points of the graph are the end points of the domain and the values of the independent variable where the derivative is zero.

As part of our study of minimal cost of production, we also derived a differentiation formula for the reciprocal function. In fact, this formula turned out to be just like the Power Rule for nonnegative integer powers: