3.5E: Exercises

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Exercise $\PageIndex{1}$

1) If c is a critical point of $f(x)$, when is there no local maximum or minimum at $c$? Explain.

2) For the function $y=x^3$, is $x=0$ both an inflection point and a local maximum/minimum?

Answer: It is not a local maximum/minimum because $f′$ does not change sign

3) For the function $y=x^3$, is $x=0$ an inflection point?

4) Is it possible for a point $c$ to be both an inflection point and a local extrema of a twice differentiable function?

Answer: No

6) Why do you need continuity for the first derivative test? Come up with an example.

7) Explain whether a concave-down function has to cross $y=0$ for some value of $x$.

Answer: False; for example, $y=\sqrt{x}$.

8) Explain whether a polynomial of degree $2$ can have an inflection point.

Answer: Under Construction.

Exercise $\PageIndex{2}$

For the following exercises, analyze the graphs of $f′$, then list all intervals where f is increasing or decreasing.

Answer: Increasing for $−2<x<−1$ and $x>2$; decreasing for $x<−2$ and $−1<x<2$

Answer: Decreasing for $x<1$, increasing for $x>1$

Answer: Decreasing for $−2<x<−1$ and $1<x<2$; increasing for $−1<x<1$ and $x<−2$ and $x>2$

Exercise $\PageIndex{3}$

For the following exercises, analyze the graphs of $f′,$ then list all intervals where

a. $f$ is increasing and decreasing and

b. the minima and maxima are located.

Answer: a. Increasing over $−2<x<−1,0<x<1,x>2$, decreasing over $x<−2, −1<x<0,1<x<2;$ b. maxima at $x=−1$ and $x=1$, minima at $x=−2$ and $x=0$ and $x=2$

Answer: a. Increasing over $x>0$, decreasing over $x<0;$ b. Minimum at $x=0$

Exercise $\PageIndex{4}$

For the following exercises, analyze the graphs of $f′$, then list all inflection points and intervals $f$ that are concave up and concave down.

Answer: Concave up on all $x$, no inflection points

Answer: Concave up on all $x$, no inflection points

Answer: Concave up for $x<0$ and $x>1$, concave down for $0<x<1$, inflection points at $x=0$ and $x=1$

Exercise $\PageIndex{5}$

For the following exercises, draw a graph that satisfies the given specifications for the domain $x=[−3,3].$ The function does not have to be continuous or differentiable.

1) $f(x)>0,f′(x)>0$ over $x>1,−3<x<0,f′(x)=0$ over $0<x<1$

2) $f′(x)>0$ over $x>2,−3<x<−1,f′(x)<0$ over $−1<x<2,f''(x)<0$ for all $x$

3) $f''(x)<0$ over $−1<x<1,f''(x)>0,−3<x<−1,1<x<3,$ local maximum at $x=0,$ local minima at $x=±2$

4) There is a local maximum at $x=2,$ local minimum at $x=1,$ and the graph is neither concave up nor concave down.

5) There are local maxima at $x=±1,$ the function is concave up for all $x$, and the function remains positive for all $x.$

Answer: Under Construction

Exercise $\PageIndex{6}$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing and

b. local minima and maxima of $f$.

1) $f(x)=sinx+sin^3x$ over −π<x<π

Answer

a. Increasing over $−\frac{π}{2}<x<\frac{π}{2},$ decreasing over $x<−π\frac{π}{2},x>\frac{π}{2}$

b. Local maximum at $x=\frac{π}{2}$; local minimum at $x=−\frac{π}{2}$

2) $f(x)=x^2+cosx$

Answer: Under Construction

Exercise $\PageIndex{7}$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing,

b. local minima and maxima of $f$,

c. intervals where $f$ is concave up and concave down, and

d. the inflection points of $f.$

1) $f(x)=x^3−4x^2+x+2$

Answer

a. Concave up for $x>\frac{4}{3},$ concave down for $x<\frac{34}{3}$

b. Inflection point at $x=\frac{4}{3}$

2) $f(x)=x^2−6x$

3) $f(x)=x^3−6x^2$

Answer

a. Increasing over $x<0$ and $x>4,$ decreasing over $0<x<4$

b. Maximum at $x=0$, minimum at $x=4$

c. Concave up for $x>2$, concave down for $x<2$

d. Infection point at $x=2$

4) $f(x)=x^4−6x^3$

5) $f(x)=x^{11}−6x^{10}$

Answer

a. Increasing over $x<0$ and $x>\frac{60}{11}$, decreasing over $0<x<\frac{60}{11}$

b. Minimum at $x=\frac{60}{11}$

c. Concave down for $x<\frac{54}{11}$, concave up for $x>\frac{54}{11}$

d. Inflection point at $x=\frac{54}{11}$

6) $f(x)=x+x^2−x^3$

7) $f(x)=x^2+x+1$

Answer

a. Increasing over $x>−\frac{1}{2}$, decreasing over $x<−\frac{1}{2}$

b. Minimum at $x=−\frac{1}{2}$

c. Concave up for all $x$

d. No inflection points

8) $f(x)=x^3+x^4$

Exercise $\PageIndex{8}$

For the following exercises, determine

a. intervals where $f$ is increasing or decreasing,

b. local minima and maxima of $f,$

c. intervals where $f$ is concave up and concave down, and

d. the inflection points of $f.$ Sketch the curve, then use a calculator to compare your answer. If you cannot determine the exact answer analytically, use a calculator.

1) $f(x)=sin(πx)−cos(πx)$ over $x=[−1,1]$

Answer

a. Increases over $−\frac{1}{4}<x<\frac{3}{4},$ decreases over $x>\frac{3}{4}$ and $x<−\frac{1}{4}$

b. Minimum at $x=−\frac{1}{4}$, maximum at $x=\frac{3}{4}$

c. Concave up for $−\frac{3}{4}<x<\frac{1}{4}$, concave down for $x<−\frac{3}{4}$ and $x>\frac{1}{4}$

d. Inflection points at $x=−\frac{3}{4},x=\frac{1}{4}$

2) $f(x)=x+sin(2x)$ over $x=[−\frac{π}{2},\frac{π}{2}]$

3) $f(x)=sinx+tanx$ over $(−\frac{π}{2},\frac{π}{2})$

Answer

a. Increasing for all $x$

b. No local minimum or maximum

c. Concave up for $x>0$, concave down for $x<0$

d. Inflection point at $x=0$

4) $f(x)=(x−2)^2(x−4)^2$

5) $f(x)=\frac{1}{1−x},x≠1$

Answer

a. Increasing for all $x$ where defined

b. No local minima or maxima

c. Concave up for $x<1$; concave down for $x>1$

d. No inflection points in domain

6) $f(x)=\frac{sinx}{x}$ over $x=[−2π,2π] [2π,0)∪(0,2π]$

7) $f(x)=sin(x)e^x$ over $x=[−π,π]$

Answer

a. Increasing over $−\frac{π}{4}<x<\frac{3π}{4}$, decreasing over $x>\frac{3π}{4},x<−\frac{π}{4}$

b. Minimum at $x=−\frac{π}{4}$, maximum at $x=\frac{3π}{4}$

c. Concave up for $−\frac{π}{2}<x<\frac{π}{2}$, concave down for $x<−\frac{π}{2},x>\frac{π}{2}$

d. Infection points at $x=±\frac{π}{2}$

8) $f(x)=lnx\sqrt{x},x>0$

9) $f(x)=\frac{1}{4}\sqrt{x}+\frac{1}{x},x>0$

Answer

a. Increasing over $x>4,$ decreasing over $0<x<4$

b. Minimum at $x=4$

c. Concave up for $0<x<8\sqrt[3]{2}$, concave down for $x>8\sqrt[3]{2}$

d. Inflection point at $x=8\sqrt[3]{2}$

10) $f(x)=\frac{e^x}{x},x≠0$

11) $f(x)=\frac{x}{ln(|x|)}$

Answer:

a. Increasing over $x>e,$ decreasing over $-1<x<e$

b. Minimum at $x=e$

c. Concave up for $(-1,0) \cup (1, \infty)\$, concave down for $(0,1)$

d. Inflection point at $x=0$

Exercise $\PageIndex{9}$

For the following exercises, interpret the sentences in terms of $f,f′,$ and $f''.$

1) The population is growing more slowly. Here $f$ is the population.

Answer:: $f>0,f′>0,f''<0$

2) A bike accelerates faster, but a car goes faster. Here $f=$ Bike’s position minus Car’s position.

3) The airplane lands smoothly. Here $f$ is the plane’s altitude.

Answer:: $f>0,f′<0,f''<0$

4) Stock prices are at their peak. Here $f$is the stock price.

5) The economy is picking up speed. Here $f$ is a measure of the economy, such as GDP.

Answer: $f>0,f′>0,f''>0$

Exercise $\PageIndex{10}$

For the following exercises, consider a third-degree polynomial $f(x),$ which has the properties f′(1)=0,f′(3)=0.

Determine whether the following statements are true or false. Justify your answer.

1) $f(x)=0$ for some $1≤x≤3$

2) $f''(x)=0$ for some $1≤x≤3$

Answer:: True, by the Mean Value Theorem

3) There is no absolute maximum at $x=3$

4) If $f(x)$ has three roots, then it has $1$ inflection point.

Answer:: True, examine derivative

5) If $f(x)$ has one inflection point, then it has three real roots.

\begin{enumerate}

\item {Consider the function $$f(x)=x^2 e^{-x}.$$

\begin{enumerate}

\item Find the interval(s) on which $f(x)$ is increasing and

\item Find the interval(s) on which $f(x)$ is decreasing and

\item Find the local (relative) maximum and local (relative) minimum (if any).

%\vspace{8in}

\end{enumerate}}

\item {Consider the function $$f(x)=3(x^2-4)^{2/3}.$$

\begin{enumerate}

\item Find the interval(s) on which $f(x)$ is increasing and

\item Find the interval(s) on which $f(x)$ is decreasing and

\item Find the local (relative) maximum and local (relative) minimum (if any).

%\vspace{8in}

\end{enumerate}}

\item Consider the function $$g(x)=10x^3+ 3x^5.$$ Find

\begin{enumerate}

\item the interval(s) on which $g(x)$ is concave up and

\item the interval(s) on which $g(x)$ is concave down and

\item inflection points ( if any).

\end{enumerate}

\item Consider the function $$g(x)=e^{-x^2/2}.$$ Find

\begin{enumerate}

\item the interval(s) on which $g(x)$ is concave up and

\item the interval(s) on which $g(x)$ is concave down and

\item inflection points ( if any).

\end{enumerate}

\item{ Find horizontal asymptote of the function $$ y=f(x)= x- \sqrt{x^2+2x-6}.$$}

%\vspace{8 in}

\item { Consider the function $$f(x)=x e^{-2x}.$$

\begin{enumerate}

\item Find the interval(s) on which $f(x)$ is increasing and

\item Find the interval(s) on which $f(x)$ is decreasing and

\item Find the local (relative) maximum and local (relative) minimum (if any).

%\vspace{8in}

\end{enumerate}}

\item { Consider the function $$f(x)=x^2 e^{-x}.$$

\begin{enumerate}

\item Find the interval(s) on which $f(x)$ is increasing and

\item Find the interval(s) on which $f(x)$ is decreasing and

\item Find the local (relative) maximum and local (relative) minimum (if any).

%\vspace{8in}

\end{enumerate}}

\item { Consider the function $$f(x)=x+\frac{1}{x}.$$

\begin{enumerate}

\item Find the interval(s) on which $f(x)$ is increasing and

\item Find the interval(s) on which $f(x)$ is decreasing and

\item Find the local (relative) maximum and local (relative) minimum (if any).

%\vspace{8in}

\end{enumerate}}

\item { Consider the function $$g(x)=5x^3+ 2x^5.$$ Find

\begin{enumerate}

\item the interval(s) on which $g(x)$ is concave up and

\item the interval(s) on which $g(x)$ is concave down and

\item inflection points ( if any).

%\vspace{8in}

\end{enumerate}}

\item { Consider the function $$g(x)=\ln(1+x^2).$$ Find

\begin{enumerate}

\item the interval(s) on which $g(x)$ is concave up and

\item the interval(s) on which $g(x)$ is concave down and

\item inflection points ( if any).

%\vspace{8in}

\end{enumerate}}

\item { Consider the function $$g(x)=(3-x^2)^2.$$ Find

\begin{enumerate}

\item the interval(s) on which $g(x)$ is concave up and

\item the interval(s) on which $g(x)$ is concave down and

\item inflection points ( if any).

%\vspace{8in}

\end{enumerate}}

\item { Consider the function

\displaystyle{f(t)=\frac{1}{1-t^2}}

b) Find vertical and horizontal asymptotes (if any).

c) Find the intervals when $f(t)$ is concave up and concave down.

}

\item {Consider the function

\displaystyle{f(t)=e^{-t^2}}

a) Find the local maxima and minima for the function $f(t)$.

b) Find the intervals when $f(t)$ is increasing and decreasing.

c) Find the intervals when $f(t)$ is concave up and down.

d) Find the inflection points for $f(t)$.

}

\item { Find all vertical and

horizontal asymptotes for the function

\displaystyle{f(x)=\frac{x^2-5x+6}{x^2-4x+3}}

Justify your work using limits.

}

\item Suppose we have the following

information, where

$$y=\frac{x}{x^2-1},\qquad y^\prime=-\frac{x^2+1}{\left(x^2-1\right)^2},

\mbox{ and }y^{\prime\prime}=\frac{2x\left(x^2+3\right)}{\left(x^2-1

\right)^3}.$$

\noindent a) Identify any asymptotes for the function $y$ using limits.

\noindent b) Find where the function $y$ is

increasing/decreasing and any relative extrema.

\noindent c) Find where the function $y$ is concave

up/down and any points of inflection.

\noindent d) Graph the function $y$.

\item Let $f(x) = x^{5/4}-x^{1/4}.$ Determine the intervals on which $f$ is increasing and decreasing and find any local extrema.

\item Let $g(x) = \ln (x^2+1).$ Determine the intervals of concavity of $g(x)$ and locate any inflection points. \item Let $f(x) = x^2e^{-16x^2}.$ Determine the intervals on which $f$ is increasing and decreasing and find any local extrema.

\item Let $g(x) = (\ln x)^2.$ Determine the intervals of concavity of $g(x)$ and locate any inflection points. \item Given a function $\displaystyle{y=f(x)= x-2tan^{-1}(x)}$, find interval(s) on which the function is concave

up and the interval(s) on which the function is concave down, and inflection point(s).

\item Let $g(x) = x-2 \tan^{-1}(x).$ Determine the intervals of concavity of $g(x)$ and locate any inflection points.

\end{enumerate}

Contributors

Gilbert Strang (MIT) and Edwin “Jed” Herman (Harvey Mudd) with many contributing authors. This content by OpenStax is licensed with a CC-BY-SA-NC 4.0 license. Download for free at http://cnx.org.

Exercise \(\PageIndex{1}\)

Exercise \(\PageIndex{2}\)

Exercise \(\PageIndex{3}\)

Exercise \(\PageIndex{4}\)

Exercise \(\PageIndex{5}\)

Exercise \(\PageIndex{6}\)

Exercise \(\PageIndex{7}\)

Exercise \(\PageIndex{8}\)

Exercise \(\PageIndex{9}\)

Exercise \(\PageIndex{10}\)

Contributors