§3.3 Using derivatives to identify extreme values

Introduction

Critical numbers and the first derivative test

The second derivative test

Consider the function $h$ given by the graph in Figure 3.1. Use the graph to answer each of the following questions.

Identify all of the values of $c$ such that $-3 \lt c \lt 3$ for which $h(c)$ is a local maximum of $h$.

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.

Determine, with justification, all critical numbers of $g$.

By developing a carefully labeled first derivative sign chart, decide whether $g$ has as a local maximum, local minimum, or neither at each critical number.

Does $g$ have a global maximum? global minimum? Justify your claims.

Sketch a possible graph of $y = g(x)$. Clearly label any local or global extrema on the graph.

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 $f$.

Identify all $x$-values where $g''(x) = 0$ or $g''(x)$ is undefined. Then, construct a second derivative sign chart for $g$ and hence state the intervals on which $g$ is concave up, as well as the intervals on which $g$ is concave down.

State the $x$-coordinates of all points of inflection of $g$.

Suppose you are given that $g'(-1.67857351) = 0$. Is there is a local maximum, local minimum, or neither (for the function $g$) at this critical number of $g$, or is it impossible to say? Why?

Assuming that $g''(x)$ is a polynomial (and that all important behavior of $g''$ is seen in the graph above), what degree polynomial do you think $g(x)$ is? Why?

Use a graphing utility to sketch the graph of $h$ for several different $k$-values, including $k = 1,3,5,10$. Plot $h(x) = x^2 + \cos(3x)$ on the axes provided. What is the smallest value of $k$ at which you think you can see (just by looking at the graph) at least one inflection point on the graph of $h$?

Explain why the graph of $h$ has no inflection points if $k \le \sqrt{2}$, but infinitely many inflection points if $k \gt \sqrt{2}$.

Explain why, no matter the value of $k$, $h$ can only have finitely many critical numbers.