Free Printable Calculus Critical Numbers, Relative and Absolute Extrema

1.

Use the diagram to classify relative and absolute extrema.

Select ALL that apply.

a)

A is an absolute minimum

b)

B is a relative maximum

c)

C is an absolute minimum

d)

D is an relative minimum

e)

E is an relative minimum

2.

Use the diagram to classify relative and absolute extrema.

Select ALL that apply.

a)

(0,0) is an absolute minimum

b)

(1,-14) is an absolute minimum

c)

(4, 40) is an absolute maximum

d)

(6, 10) is an relative minimum

e)

(7, 65) is an absolute maximum

3.

Find the critical number of f(x) = 4x² – 8x + 1

a)

x = 2

b)

x = 0

c)

x = 8

d)

x = 1

4.

Find all the critical numbers of

f(x) = x³ – 6x² + 10.

Select ALL that apply

a)

x = 4

b)

x = 6

c)

x = -2

d)

x = 0

5.

Find all the critical numbers of

f(x) = 2x³ – 6x² + 12

Select ALL that apply.

a)

x= 4

b)

x= 0

c)

x= -2

d)

x= 2

6.

$f\left(x\right)=\frac{2}{3}x^3-18x^2-80x+17$

Find all the critical numbers of f(x) above
Select ALL that apply.

a)

x= -10

b)

x= 20

c)

x= -8

d)

x= -2

7.

$f\left(x\right)=\frac{1}{3}x^3+3x^2+8x-5$
Find the critical numbers of f(x). Select ALL that apply

a)

x=2

b)

x=0

c)

x= -4

d)

x= -2

8.

Find all the critical numbers of

f(x) = 6x⁴ – 16x³ – 180x² + 456.

Select ALL that apply

a)

x= -3

b)

x= 5

c)

x= 0

d)

x = 7

e)

x = 1

9.

Find all the critical numbers of f(x).

f\left(x\right)=\frac{1}{4}x^4-\frac{5}{3}x^3-7x^2+13

Select ALL that apply.

a)

x= 7

b)

x= 5

c)

x= 0

d)

x = -2

e)

x = 14

10.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=6\sin x-3\sqrt{2}x$ . Select ALL that apply

a)

$x=\frac{7\pi}{4}$

b)

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

c)

$x=\frac{\pi}{4}$

d)

$x=\frac{5\pi}{4}$

e)

$x=\pi$

11.

Given $0\le x\le2\pi$ , Find ALL the critical numbers in

f\left(x\right)=\sin x

a)

$x=0$

b)

$x=\pi$

c)

$x=2\pi$

d)

$x=\frac{3\pi}{2}$

e)

$x=\frac{\pi}{2}$

12.

Given $0\le x\le2\pi$ , Find ALL the critical numbers in

f\left(x\right)=\cos x

. Select ALL that apply.

a)

$x=0$

b)

$x=\pi$

c)

$x=2\pi$

d)

$x=\frac{3\pi}{2}$

e)

$x=\frac{\pi}{2}$

13.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=10\sin x+5\sqrt{2}x$ . Select ALL that apply

a)

$x=\frac{7\pi}{4}$

b)

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

c)

$x=\frac{\pi}{4}$

d)

$x=\frac{5\pi}{4}$

e)

$x=\pi$

14.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=8\cos x-4\sqrt{2}x$ . Select ALL that apply

a)

$x=\frac{7\pi}{4}$

b)

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

c)

$x=\frac{\pi}{4}$

d)

$x=\frac{5\pi}{4}$

e)

$x=\pi$

15.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=18\cos x+9\sqrt{2}x$ . Select ALL that apply

a)

$x=\frac{7\pi}{4}$

b)

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

c)

$x=\frac{\pi}{4}$

d)

$x=\frac{5\pi}{4}$

e)

$x=\pi$

16.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=4\sin x-2\sqrt{3}x$ . Select ALL that apply

a)

$x=\frac{5\pi}{6}$

b)

$x=\frac{7\pi}{6}$

c)

$x=\frac{\pi}{6}$

d)

$x=\frac{11\pi}{6}$

e)

$x=\frac{5\pi}{3}$

17.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=-6\cos x+3\sqrt{3}x$ . Select ALL that apply

a)

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

b)

$x=\frac{2\pi}{3}$

c)

$x=\frac{\pi}{3}$

d)

$x=\frac{5\pi}{3}$

e)

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

18.

Given $0\le x\le2\pi$ , Find ALL the critical numbers in

f\left(x\right)=2\sin x-2x

. Select ALL that apply.

a)

$x=0$

b)

$x=\pi$

c)

$x=2\pi$

d)

$x=\frac{3\pi}{2}$

e)

$x=\frac{\pi}{2}$

19.

Given $0\le x\le2\pi$ , Find ALL the critical numbers in

f\left(x\right)=5\cos x-5x

. Select ALL that apply.

a)

$x=0$

b)

$x=\pi$

c)

$x=2\pi$

d)

$x=\frac{3\pi}{2}$

e)

$x=\frac{\pi}{2}$

20.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=6\sin x-3x$ . Select ALL that apply

a)

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

b)

$x=\frac{2\pi}{3}$

c)

$x=\frac{\pi}{3}$

d)

$x=\frac{5\pi}{3}$

e)

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

21.

Given $\left(0\le x\le2\pi\right)$ Find ALL the critical numbers in $f\left(x\right)=14\cos x+7x$ . Select ALL that apply

a)

$x=\frac{5\pi}{6}$

b)

$x=\frac{7\pi}{6}$

c)

$x=\frac{\pi}{6}$

d)

$x=\frac{11\pi}{6}$

e)

$x=\frac{5\pi}{3}$

22.

Find the critical number of

f\left(x\right)=2e^x-2ex

a)

x = 2

b)

x = 0

c)

x = 1

d)

x = -1

e)

x = e

23.

Find the critical number of

f\left(x\right)=2e^x-2x

a)

x = 2

b)

x = 0

c)

x = 1

d)

x = -1

e)

x = e

24.

Find the critical number of

f\left(x\right)=2\ln x-x

a)

x = 2

b)

x = 0

c)

x = 1

d)

x = -1

e)

$x=\frac{1}{2}$

25.

Find the critical number of

f\left(x\right)=2\ln x-4x

a)

x = 2

b)

x = 0

c)

x = 1

d)

x = -1

e)

$x=\frac{1}{2}$

26.

Given $0\le x\le2\pi$ , Find ALL the critical numbers in

f\left(x\right)=3\sin x+3\cos x

. Select ALL that apply.

a)

$x=\frac{\pi}{4}$

b)

$x=\pi$

c)

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

d)

$x=\frac{5\pi}{4}$

e)

$x=\frac{7\pi}{4}$

Calculus Critical Numbers, Relative and Absolute Extrema

Total questions: 26

Worksheet time: 7hrs 30mins

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