Chapter 4 - Section 4.3 - How Derivatives Affect the Shape of a Graph - 4.3 Exercises - Page 300: 4

(a) Let $c$ be a critical number of a continuous function $f$ If $f'$ changes from positive to negative at $c$, then the function $f$ has a local maximum at $c$ If $f'$ changes from negative to positive at $c$, then the function $f$ has a local minimum at $c$ If $f'$ is positive to the left of $c$ and to the right of $c$, or if $f'$ is negative to the left of $c$ and to the right of $c$, then $f$ does not have a local maximum or a local minimum at $c$ (b) Let $f''$ be continuous near $c$. If $f'(c) = 0$ and $f''(c) \gt 0$, then $f$ has a local minimum at $c$ If $f'(c) = 0$ and $f''(c) \lt 0$, then $f$ has a local maximum at $c$ When $f''(c) = 0$, then the Second Derivative Test is inconclusive. In that case, we must use the First Derivative Test.

(a) The First Derivative Test can be used to find local maxima and local minima. The First Derivative Test can be stated as follows: Let $c$ be a critical number of a continuous function $f$ If $f'$ changes from positive to negative at $c$, then the function $f$ has a local maximum at $c$ If $f'$ changes from negative to positive at $c$, then the function $f$ has a local minimum at $c$ If $f'$ is positive to the left of $c$ and to the right of $c$, or if $f'$ is negative to the left of $c$ and to the right of $c$, then $f$ does not have a local maximum or a local minimum at $c$ (b) The Second Derivative Test can be used to find local maxima and local minima. The Second Derivative Test can be stated as follows: Let $f''$ be continuous near $c$. If $f'(c) = 0$ and $f''(c) \gt 0$, then $f$ has a local minimum at $c$ If $f'(c) = 0$ and $f''(c) \lt 0$, then $f$ has a local maximum at $c$ When $f''(c) = 0$, then the Second Derivative Test is inconclusive. In that case, we must use the First Derivative Test.