4.5E: Exercises

 

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.

1)

Answer

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

2)

3)

Answer

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

 

4)

5)

 

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.

 

1)

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 $x=2$

 

2)

3)

Answer

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

 

4)

 

 

 

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.

1)

 

Answer

Concave up on all $x$, no inflection points

2)

3)

Answer

Concave up on all $x$, no inflection points

 

4)

5)

 

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

2)Decreasing for \(−1<x<2\) ; increasing for \(−3<x<−1\) and $x>2$; Concave down for all $x$.

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) $f(x)=sin(\pi x)−cos(\pi x)$ 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 $x=−\frac{3}{4},x=\frac{1}{4}$

Solution

Since $f(x)=sin(\pi x)−cos(\pi x)$ over $x=[−1,1]$, $f'(x)=\pi cos(\pi x)+\pi sin(\pi x)$.Critical points:  $f'(x)=\pi cos(\pi x)+\pi sin(\pi x)=0 \implies \pi cos(\pi x)=-\pi sin(\pi x) \implies cos(\pi x)=-sin(\pi x) \implies \pi x= \dfrac{3\pi}{4}, \dfrac{-\pi}{4} \implies x= \dfrac{3}{4}, \dfrac{-1}{4}.$ Note that $\pi x\in [-\pi , \pi]$, not $[0 , 2\pi].$

Thus, 

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 $x=\frac{3}{4}$.

Concavity:  $f'(x)=\pi cos(\pi x)+\pi sin(\pi x) \implies f''(x)=-\pi^2 sin(\pi x)+\pi^2 cos(\pi x)$. Now solve  $f''(x)=-\pi^2 sin(\pi x)+\pi^2 cos(\pi x)=0.$ Thus   $cos(\pi x)=sin(\pi x)$. Which implies, $\pi x= \dfrac{-3\pi}{4}, \dfrac{\pi}{4} \implies x= \dfrac{-3}{4}, \dfrac{1}{4}.$

Thus,

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 $x=8\sqrt[3]{2}$

Solution

$f(x)=\dfrac{1}{4}\sqrt{x}+\dfrac{1}{x},x>0. f'(x)=\dfrac{1}{8\sqrt{x}}+\dfrac{-1}{x^2}=\dfrac{x^2- 8\sqrt{x}}{8x^2\sqrt{x}}=\dfrac{x^{3/2}- 8}{8x^2},$ since $x >0.$  Critical points: $x^{3/2}- 8=0 \implies x^{3/2}= 8 \implies x=(8)^{2/3} =(2^3)^{2/3} \implies x=4.$ We will create sign chart using test points:

	$(0,4)$ Test point $x=2$	$(4,\infty)$ Test point $x=8$
$x^{3/2}- 8$	$-$	$+$
$8x^2$	$+$	$+$
Sign of $f'$	$-$	$+$

Thus $f$ increasing over $x>4,$ decreasing over $0<x<4$ and has a local minimum of  $f(4)=\dfrac{1}{4}\sqrt{4}+\dfrac{1}{4}=\dfrac{3}{4}$ at $x=4$.

Concavity: $f'(x)=\dfrac{1}{8\sqrt{x}}+\dfrac{-1}{x^2}, f''(x)=\dfrac{-1}{16x^{3/2}}+\dfrac{2}{x^3}= \dfrac{-x^{3/2}-32}{16x^3}.$ Now solve $x^{3/2}-32=0 \implies x^{3/2}=32 \implies x=32^{2/3}=2^{10/3}=8\sqrt[3]{2}.$ 

Using the table below: 

$f$ is concave up for $0<x<8\sqrt[3]{2}$, concave down for $x>8\sqrt[3]{2}$ and has an inflection point at $x=8\sqrt[3]{2}$.

	$(0,8\sqrt[3]{2})$ Test point $x=4$	$(8\sqrt[3]{2},\infty)$ Test point $x=16$
$x^{3/2}- 32$	$-$	$+$
$16x^3$	$+$	$+$
Sign of $f'$	$-$	$+$

 

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.

Exercise $\PageIndex{11}$

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.  $f(x)=x^2 e^{-x}$

2.  $f(x)=3(x^2-4)^{2/3}$

3.  $f(x)=10x^3+ 3x^5$

4.  $f(x)=e^{-x^2/2}$

5. $f(x)=x e^{-2x}$

6. $f(x)=x+\frac{1}{x}$

7. $f(x) =5x^3+ 2x^5$

8. $f(x)=\ln(1+x^2)$

9. $f(x)=(3-x^2)^2$

10. $f(t)=\dfrac{1}{1-t^2}$

11. $f(t)=e^{-t^2}$

12. $f(x) = x^{5/4}-x^{1/4}$

13. $f(x) = \ln (x^2+1)$

14. $f(x) = (\ln x)^2$

15. $f(x) = x-2 \tan^{-1}(x)$

 

Exercise $\PageIndex{12}$

Suppose we have the following information where

$y=\dfrac{x}{x^2-1}, y'=-\dfrac{x^2+1}{\left(x^2-1\right)^2},$ and $y''=\dfrac{2x\left(x^2+3\right)}{\left(x^2-1\right)^3}.$

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

b) Find where the function $y$ is increasing/decreasing and any relative extrema.

c) Find where the function $y$ is concave up/down and any points of inflection.

 

Contributors and Attributions

• 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.

• Exercises 11 and 12 are by Pamini Thangarajah