4.4E: Exercises for Section 4.4
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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=x3, 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=x3, is x=0 an inflection point?
4) Is it possible for a point c to be both an inflection point and a local extremum of a twice differentiable function?
- Answer
- No
5) Why do you need continuity for the first derivative test? Come up with an example.
6) Explain whether a concave-down function has to cross y=0 for some value of x.
- Answer
- False; for example, y=√x.
7) Explain whether a polynomial of degree 2 can have an inflection point.
In exercises 8 - 12, analyze the graphs of f′, then list all intervals where f is increasing or decreasing.
8)
- Answer
- Increasing for −2<x<−1 and x>2;
Decreasing for x<−2 and −1<x<2
9)
10)
- Answer
- Decreasing for x<1,
Increasing for x>1
11)
12)
- Answer
- Decreasing for −2<x<−1 and 1<x<2;
Increasing for −1<x<1 and x<−2 and x>2
In exercises 13 - 17, analyze the graphs of f′, then list all intervals where
a. f is increasing and decreasing and
b. the minima and maxima are located.
13)
14)
- 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
15)
16)
- Answer
- a. Increasing over x>0, Decreasing over x<0;
b. Minimum at x=0
17)
In exercises 18 - 22, analyze the graphs of f′, then list all inflection points and intervals f that are concave up and concave down.
18)
- Answer
- Concave up for all x,
No inflection points
19)
20)
- Answer
- Concave up for all x,
No inflection points
21)
22)
- Answer
- Concave up for x<0 and x>1,
Concave down for 0<x<1,
Inflection points at x=0 and x=1
For exercises 23 - 27, draw a graph that satisfies the given specifications for the domain x=[−3,3]. The function does not have to be continuous or differentiable.
23) f(x)>0,f′(x)>0 over x>1,−3<x<0,f′(x)=0 over 0<x<1
24) f′(x)>0 over x>2,−3<x<−1,f′(x)<0 over −1<x<2,f″(x)<0 for all x
- Answer
- Answers will vary
25) 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
26) There is a local maximum at x=2, local minimum at x=1, and the graph is neither concave up nor concave down.
- Answer
- Answers will vary
27) There are local maxima at x=±1, the function is concave up for all x, and the function remains positive for all x.
For the following exercises, determine
a. intervals where f is increasing or decreasing and
b. local minima and maxima of f.
28) f(x)=sinx+sin3x over −π<x<π
- Answer
-
a. Increasing over −π2<x<π2, decreasing over x<−π2,x>π2
b. Local maximum at x=π2; local minimum at x=−π2
29) f(x)=x2+cosx
For exercise 30, determine
a. intervals where f is concave up or concave down, and
b. the inflection points of f.
30) f(x)=x3−4x2+x+2
- Answer
-
a. Concave up for x>43, concave down for x<43
b. Inflection point at x=43
For exercises 31 - 37, 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.
31) f(x)=x2−6x
32) f(x)=x3−6x2
- 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. Inflection point at x=2
33) f(x)=x4−6x3
34) f(x)=x11−6x10
- Answer
- a. Increasing over x<0 and x>6011, decreasing over 0<x<6011
b. Maximum at x=0, minimum at x=6011
c. Concave down for x<5411, concave up for x>5411
d. Inflection point at x=5411
35) f(x)=x+x2−x3
36) f(x)=x2+x+1
- Answer
- a. Increasing over x>−12, decreasing over x<−12
b. Minimum at x=−12
c. Concave up for all x
d. No inflection points
37) f(x)=x3+x4
For exercises 38 - 47, 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.
38) [T] f(x)=sin(πx)−cos(πx) over x=[−1,1]
- Answer
- a. Increases over −14<x<34, decreases over x>34 and x<−14
b. Minimum at x=−14, maximum at x=34
c. Concave up for −34<x<14, concave down for x<−34 and x>14
d. Inflection points at x=−34,x=14
39) [T] f(x)=x+sin(2x) over x=[−π2,π2]
40) [T] f(x)=sinx+tanx over (−π2,π2)
- 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
41) [T] f(x)=(x−2)2(x−4)2
42) [T] f(x)=11−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
43) [T] f(x)=sinxx over x=[−2π,0)∪(0,2π]
44) f(x)=sin(x)ex over x=[−π,π]
- Answer
- a. Increasing over −π4<x<3π4, decreasing over x>3π4,x<−π4
b. Minimum at x=−π4, maximum at x=3π4
c. Concave up for −π2<x<π2, concave down for x<−π2,x>π2
d. Inflection points at x=±π2
45) f(x)=lnx√x,x>0
46) f(x)=14√x+1x,x>0
- Answer
- a. Increasing over x>4, decreasing over 0<x<4
b. Minimum at x=4
c. Concave up for 0<x<83√2, concave down for x>83√2
d. Inflection point at x=83√2
47) f(x)=exx,x≠0
In exercises 48 - 52, interpret the sentences in terms of f, f′, and f″.
48) The population is growing more slowly. Here f is the population.
- Answer
- f>0,f′>0,f″<0
49) A bike accelerates faster, but a car goes faster. Here f= Bike’s position minus Car’s position.
50) The airplane lands smoothly. Here f is the plane’s altitude.
- Answer
- f>0,f′<0,f″>0
51) Stock prices are at their peak. Here fis the stock price.
52) The economy is picking up speed. Here f is a measure of the economy, such as GDP.
- Answer
- f>0,f′>0,f″>0
For exercises 53 - 57, consider a third-degree polynomial f(x), which has the properties f′(1)=0 and f′(3)=0.
Determine whether the following statements are true or false. Justify your answer.
53) f(x)=0 for some 1≤x≤3.
54) f″(x)=0 for some 1≤x≤3.
- Answer
- True, by the Mean Value Theorem
55) There is no absolute maximum at x=3.
56) If f(x) has three roots, then it has 1 inflection point.
- Answer
- True, examine derivative
57) If f(x) has one inflection point, then it has three real roots.