Chapter 8

Let \(f(x)=e^{x} .\) Show that the hypotheses of the Mean Value Theorem are satisfied on the interval [0,1] and find all values of \(c\) that satisfy the conclusion of the theorem.

Determine the intervals in which the graph of \(f(x)=\frac{x^{2}+9}{x^{2}-25}\) is

Given \(f(x)=x+\sin x 0 \leq x \leq 2 \pi,\) find all points of inflection of \(f\).

Show that the absolute minimum of \(f(x)=\sqrt{25-x^{2}}\) on [-5,5] is 0 and the absolute maximum is 5.

If \(f^{\prime \prime}(x)=x^{2}(x+3)(x-5),\) find the values of \(x\) at which the graph of \(f\) has a change of concavity.

The graph of \(f^{\prime}\) on [-3,3] is shown in Figure \(8.6-3\). Find the values of \(x\) on [-3,3] such that (a) \(f\) is increasing and \((b) f\) is concave downward.

Sketch the graphs of the following functions indicating any relative and absolute extrema, points of inflection, intervals on which the function is increasing, decreasing, concave upward, or concave downward. $$ f(x)=x^{4}-x^{2} $$

Sketch the graphs of the following functions indicating any relative and absolute extrema, points of inflection, intervals on which the function is increasing, decreasing, concave upward, or concave downward. $$ f(x)=\frac{x+4}{x-4} $$

A function \(f\) is continuous on the interval [-2,5] with \(f(-2)=10\) and \(f\) \((5)=6\) and the following properties: $$ \begin{array}{c|c|c|c|c|c} \text { INTERVALS } & (-2,1) & X=1 & (1,3) & X=3 & (3,5) \\ \hline f^{\prime} & \+ & 0 & \- & \text { undefined } & \+ \\ \hline f^{\prime \prime} & \- & 0 & \- & \text { undefined } & + \end{array} $$ (a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find where \(f\) has its absolute extrema. (c) Find where \(f\) has points of inflection. (d) Find the intervals where \(f\) is concave upward or downward. (e) Sketch a possible graph of \(f\).

If \(f(x)=\left|x^{2}-6 x-7\right|,\) which of the following statements about \(f\) are true? I. \(f\) has a relative maximum at \(x=3\). II. \(f\) is differentiable at \(x=7\). III. \(f\) has a point of inflection at \(x=-1\).

How many points of inflection does the graph of \(y=\cos \left(x^{2}\right)\) have on the interval \([-\pi, \pi] ?\)

Sketch the graphs of the following functions indicating any relative extrema, points of inflection, asymptotes, and intervals where the function is increasing, decreasing, concave upward, or concave downward. $$ f(x)=3 e^{-x^{2} / 2} $$

Sketch the graphs of the following functions indicating any relative extrema, points of inflection, asymptotes, and intervals where the function is increasing, decreasing, concave upward, or concave downward. $$ f(x)=\cos x \sin ^{2} x[0,2 \pi] $$

Find \(\frac{d y}{d x}\) if \(\left(x^{2}+y^{2}\right)^{2}=10 x y\).

Evaluate \(\lim _{x \rightarrow 0} \frac{\sqrt{x+9}-3}{x}\).

Find \(\frac{d^{2} y}{d x^{2}}\) if \(y=\cos (2 x)+3 x^{2}-1\).

(Calculator) Determine the value of \(k\) such that the function \(f(x)=\left\\{\begin{array}{ll}x^{2}-1, & x \leq 1 \\ 2 x+k, & x>1\end{array}\right.\) is continuous for all real numbers.

A function \(f\) is continuous on the interval [-1,4] with \(f(-1)=0\) and \(f\) \((4)=2\) and the following properties: $$ \begin{array}{c|c|c|c|c|c} \text { INTERVALS } & (-1,0) & X=0 & (0,2) & X=2 & (2,4) \\ \hline f^{\prime} & \+ & \text { undefined } & \+ & 0 & \- \\ \hline f^{\prime \prime} & \+ & \text { undefined } & \- & 0 & - \end{array} $$ (a) Find the intervals on which \(f\) is increasing or decreasing. (b) Find where \(f\) has its absolute extrema. (c) Find where \(f\) has points of inflection. (d) Find intervals on which \(f\) is concave upward or downward. (e) Sketch a possible graph of \(f\).