Practice Problems

In problem 1, each quotation is a statement about a quantitity of something changing over time. Let $\mathrm{f}(\mathrm{t})$ represent the quantity at time $\mathrm{t}$. For each quotation, tell what $\mathrm{f}$ represents and whether the first and second derivatives of $\mathrm{f}$ are positive or negative.

1. (a) "Unemployment rose again, but the rate of increase is smaller than last month".
(b) "Our profits declined again, but at a slower rate than last month".
(c) "The population is still rising and at a faster rate than last year".

3. Sketch the graphs of functions which are defined and concave up everywhere and which have
(a) no roots.
(b) exactly 1 root.
(c) exactly 2 roots.
(d) exactly 3 roots.

5. On which intervals is the function in Fig. 12
(a) concave up?
(b) concave down?

Fig. 12

In problems 7 and 9, a function and values of $x$ so that $f^{\prime}(x)=0$ are given. Use the Second Derivative Test to determine whether each point $(x, f(x))$ is a local maximum, a local minimum or neither

7. $\mathrm{g}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-9 \mathrm{x}+7, \mathrm{x}=-1,3$

9. $\mathrm{f}(\mathrm{x})=\sin ^{5}(\mathrm{x}), \quad \mathrm{x}=\pi / 2, \pi, 3 \pi / 2$

11. At which labeled values of $\mathrm{x}$ in Fig. 13 is the point $(\mathrm{x}, \mathrm{f}(\mathrm{x}))$ an inflection point?

Fig. 13

13. How many inflection points can a
(a) quadratic polynomial have?
(b) cubic polynomial have?
(c) polynomial of degree n have?

15. Fill in the table with "$+$", "$-$", or "$0$" for the function in Fig. 16

$\begin{array}{l|l|l|l}x & g(x) & g^{\prime}(x) & g^{\prime \prime}(x) \\\hline 0 & & & \\1 & & & \\2 & & & \\3 & & &\end{array}$

Fig. 16

17. Some people like to think of a concave up graph as one which will "hold water" and of a concave down graph as one which will "spill water". That description is accurate for a concave down graph, but it can fail for a concave up graph. Sketch the graph of a function which is concave up on an interval, but which will not "hold water".

Source: Dale Hoffman, https://s3.amazonaws.com/saylordotorg-resources/wwwresources/site/wp-content/uploads/2012/12/MA005-4.4-Second-Derivative.pdf
This work is licensed under a Creative Commons Attribution 3.0 License.

1. (a) $\mathrm{f}(\mathrm{t})=$ number of workers unemployed at time $t$. $\mathrm{f}^{\prime}(\mathrm{t}) > 0$ and $\mathrm{f}^{\prime \prime}(\mathrm{t}) < 0$.
(b) $\mathrm{f}(\mathrm{t})=$ profit at time $t$. $\mathrm{f}^{\prime}(\mathrm{t}) < 0$ and $\mathrm{f}^{\prime \prime}(\mathrm{t}) > 0$.
(c) $f(t)=$ population at time $t$. $(f^{\prime}(t) > 0$ and $f^{\prime \prime}(t) > 0$.

3. See Fig. 3.4P3.

Fig. 3.4P3.

5. (a) Concave up on $(0,2),(2,3+),(6,9)$. Concave down on $(3+, 6)$. (A small technical note: we have defined concavity only at points where the function is differentiable, so we exclude the endpoints and points where the function is not differentiable from the intervals of concave up and concave down.)

7. $\mathrm{g}(\mathrm{x})=\mathrm{x}^{3}-3 \mathrm{x}^{2}-9 \mathrm{x}+7$. $\mathrm{g}^{\prime \prime} (\mathrm{x})=6 \mathrm{x}-6$. $\mathrm{g}^{\prime \prime} (-1) < 0$ so $(-1,12)$ is a local maximum. $\mathrm{g}^{\prime \prime} (3) > 0$ so $(3,-20)$ is a local minimum.

9. $\mathrm{f}(\mathrm{x})=\sin ^{5}(\mathrm{x}) \cdot \mathrm{f}^{\prime \prime} (\mathrm{x})=5\left\{-\sin ^{5}(\mathrm{x})+4 \sin ^{3}(\mathrm{x}) \cdot \cos ^{2}(\mathrm{x})\right\}$. $\mathrm{f}^{\prime \prime} (\pi / 2) < 0$ so $(\pi / 2,1)$ is a local maximum. $f^{\prime \prime} (3 \pi / 2) > 0$ so $(3 \pi / 2,-1)$ is a local minimum.
$f^{\prime \prime} (\pi)=0$ and $f$ changes concavity at $x=\pi$ so $(\pi, 0)$ is an inflection point.

11. $\mathrm{d}$ and $\mathrm{e}$.

13. (a) $0$
(b) at most $1$
(c) at most $\mathrm{n}-2$.

15.

$\begin{array}{l|l|l|l}x & g(x) & g^{\prime}(x) & g^{\prime \prime}(x) \\\hline 0 & - & + & + \\1 & + & 0 & - \\2 & - & - & + \\3 & 0 & + & +\end{array}$

17. See Fig. 3.4P17.

Fig. 3.4P17.