Problem

Show that there is a point at which the derivative of the function p(x) = e^x – 3x is equal to zero.

Solution

It appears from the graph that somewhere around x = 1 the slope of the tangent line to the curve is horizontal, but this is not proof.

Let's find the equation for the derivative of p.

This is a continuous function, therefore the Intermediate Value Theorem applies. Since there are points at which p'(x) is positive

and negative

there must be a value c at which p'(c) = 0.

This is enough to show that a root of the derivative exists, but we can go further and find this value of c by solving the equation p'(c) = 0.

Here's the graph again with the point (c, p(c)) included.