§3.3 Using derivatives to identify extreme values

Introduction

Critical numbers and the first derivative test

The second derivative test

Let $f$ be a function whose derivative is given by the formula $f'(x) = e^{-2x}(3-x)(x+1)^2$. Determine all critical numbers of $f$ and decide whether a relative maximum, relative minimum, or neither occurs at each.

Let $f(x)$ be a function whose first derivative is $f'(x) = 3x^4 - 9x^2$. Construct both first and second derivative sign charts for $f$, fully discuss where $f$ is increasing and decreasing and concave up and concave down, identify all relative extreme values, and sketch a possible graph of $f$.

Suppose $g'(x) = \frac{(x+4)(x-2)}{x^2+1}$. The critical numbers of $g$ are

For that same $g$, the sign chart of $g'$ reads positive on $x \lt -4$, negative on $-4 \lt x \lt 2$, and positive on $x \gt 2$. At $x = -4$ the function $g$ has

Suppose $g'(-1.67857351) = 0$ and the graph of $g''$ shows $g''(-1.67857351) \gt 0$. Then at that input $g$ has

For $g'(x) = \frac{(x+4)(x-2)}{x^2+1}$, compute $g'(2)$.

The graph of $g''$ shows $g'' \gt 0$ for $x \lt -1$ and $g'' \lt 0$ for $x \gt -1$. State the $x$-coordinate of the inflection point of $g$.

A critical number of $f$ where $f'$ does not change sign can fail to be the location of any local maximum or local minimum.