Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 4

a.) What is the first derivative test?
b.) What is the second derivative test? Under what circumstances is it inconclusive? What must be done if it fails?

a.) The first derivative test is used to determine if the critical number is either a local maximum or a local minimum. Here's how it works.
The first derivative test. Suppose that $c$ is a critical number of a continuous function $f$.
If $f'$ changes from positive to negative at $c$, then $f$ has a local maximum.
If $f'$ changes from negative to positive at $c$, then $f$ has a local minimum.
If $f'$ does not change sign at $c$, then $f$ has no local maximum or local minimum at $c$.

b.) The second derivative test is used to determine the concavity of the function on the given interval. If $f''(x) > 0$ for a specific interval, then the graph of $f$ is concave upward. On the other hand, if $f''(x) < 0$ for a specific interval, then the graph of $f$ is concave downward. The second derivative test is inconclusive when $f''(x) = 0$. In that case we will use the first derivative test.