4.3E: Exercises for Section 4.3

 

For exercises 1 - 22,

a. find intervals where \(f\) is increasing or decreasing,

b. state local minima and maxima of \(f\),

c. find intervals where \(f\) is concave up and concave down, 

d. state the inflection points of \(f\),

e. sketch the graph of \(f\), and

f. use your graphing calculator to verify your graph.

1) \(f(x)=x^2−6x\)

2) \(f(x)=x^3−6x^2+9x\)

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 \(x=2\)

3) \(f(x)=x^4−6x^3\)

4) \(f(x)=x^{11}−6x^{10}\)

Answer

a. Increasing over \(x<0\) and \(x>\frac{60}{11}\), decreasing over \(0<x<\frac{60}{11}\)
b. Maximum at \(x=0\), 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 \(x=\frac{54}{11}\)

5) \(f(x)=x+x^2−x^3\)

6) \(f(x)=x^2+x+1\)

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

7) \(f(x)=x^3+x^4\)

8) \(f(x)=(x−2)^2(x−4)^2\)

9) \(f(x)=(x-6)(x+6)^2\)

10) \(f(x)=\frac{2x^2-3x-2}{x^2-1}\)

11)  \(f(x)=\frac{x^2+3x+2}{x-2}\)

12)  \(f(x)=\frac{x^2}{x-2}\)

13)  \(f(t)=\frac{3t}{t^2-1}\)

14)  \(f(x)=\frac{x^2+8}{2x-1}\)

15)  \(f(x)=\sqrt[3]{2x-5}\)

16)  \(f(x)=(1-x)^{\frac{2}{3}}\)

17)  \(f(x)=x^{3/2}-5x^{1/2}\)

18)  \(f(x)=\frac{4}{x^4}-\frac{8}{x^2}\)

Answer

a. Increasing over \((-1,0)\cup(1,\infty)\), decreasing over \((-\infty,-1)\cup(0,1)\)
b. Minima at \(x=-1\) and \(x=1\)
c. Concave up over \((-\sqrt{\frac{5}{3}},0)\cup(0,\sqrt{\frac{5}{3}})\), concave down over \((-\infty,-\sqrt{\frac{5}{3}})\cup(\sqrt{\frac{5}{3}},\infty)\)

d. Inflection points at \(x=-\sqrt{\frac{5}{3}}\) and \(x=-\sqrt{\frac{5}{3}}\).

19)  \(f(t)=t\sqrt{6-t}\)

20)  \(f(x)=x^{1/2}+x^{-1/2}\) on \([0,\infty]\)

21)  \(f(x)=\sin{x}+x\) on \([0,2\pi]\)

22)  \(f(x)=2\cos{x}-x\) on \([-2\pi,2\pi]\)

 

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