Exercise Chapter 3

1. For the function y=f (x ) below, find all relative maximum points and minimum points by applying the first derivative test. Then, determine the intervals where f (x) is increasing and decreasing.i) f ( x )=x2−2x−24

ii) f ( x )=x3−3x

2. Find (a) the intervals of increase or decrease, (b) the local maximum and minimum values, (c) the intervals of concavity, and (d) the inflection points. (e) sketch the graph.i) f ( x )=2x3−3 x2−12 x

ii) f ( x )=x 4−6 x2

iii) f ( x )=3 x5−5 x3+3

3. Sketch a graph of a rational function and label the coordinates of the stationary points and inflection points. Show the horizontal and vertical asymptotes and label them with their equations. Label points, if any, where the graph crosses horizontal asymptotes.

i) y= xx−1

ii) y= x

x2+9

iii) y= x2

x+8iv) y=x2+4

v) y=2 x3+x2+1x2+1

2i 2ii 2iii 3i

3ii 3iii 3iv 3v

1. Find the critical numbers and the relative extrema for the functions, if any:

(a) y=x3−3 x+3 {ans: x=-1, 1, rel max (-1,5), rel min (1,1) }

(b) y=2x−3 x23

{ans: none}

(c) y=x √8−x2 (ans: x=-2,2, rel max at (2,4), rel min at (-2,-4)}

(d) y= x

2−3x−2 {ans: x=1, 2, 3, rel max (1,2), rel min (3,6) }

(e) y=|x2−1| {ans: x=0, rel max (0,1)

(f) y=x13 (x+3 )

23

{ans: x=0, -3, no rel extrema}

2. For each of the given function;

i) find the x and y intercepts (if any).ii) all the asymptotes (if any)iii) the interval of increase and decreaseiv) local maximum / local minimumv) interval of concavityvi) inflection point (if any)vii) sketch the function completely.

(a) f ( x )=9 x3−4 x 4

(b) f ( x )=−2x3+6 x2−3

f(x)=9x^3-4x^4

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y f(x)=-2x^3+6x^2-3

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

(c) f ( x )= 1+2 x

(1−x )2 (d) f ( x )= x

3−1x2−9

f(x)=((x^3)-1)/((x^2)-9)

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

3. A curve has the equation y=x3+ax2+bx+c . The curve cuts the y-axis at

y=−13and has stationary points at x=−1 and x=−7

3 .

(a) Find the values of a, b and c. {ans: a=5, b=7, c=-13}

(b) Find the inflection points {ans: x=-5/3}

f(x)=(1+2x)/(1-x)^2

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-9

-8

-7

-6

-5

-4

-3

-2

-1

1

2

3

4

5

6

7

8

9

x

y

(c) Sketch the graph of y.f(x)=x^3 + 5 x^2 +7x -13

-9 -8 -7 -6 -5 -4 -3 -2 -1 1 2 3 4 5 6 7 8 9

-25

-20

-15

-10

-5

5

10

15

20

25

x

y

Question

1. Find the critical points for the curvea. y=x3+7 x2−5 x+2b. f ( x )=√x2−9c. x2+2 y2−2x+8 y−9=0d. y=35x

53−6 x

23

e. y=( x2−16 )23

2. Given y=3 x4−4 x3. Find if exist, the maximum and minimum points using the first derivative test.3. A curve is given by the function f ( x )= 3√x−3.a. Find the first and second derivative of f .b. Find the coordinates ot the critical point(s).c. Determine the nature of the points whether they are maximum, minimum or point of inflection.4. For the function f ( x )=x 4−8 x2 , finda. the stationary points.

b. the intervals where f is increasing or decreasing.c. the relative maximum and minimum points.d. the intervals where f is concave upwards and f is concave downwards.e. The points of inflection.Hence sketch the graph of f (x).5. Sketch the graph of a. y=4 x2+ 1

xb. y= x2+1x2−9c. y= 2x

9−x2

d. y= 2 x2

x2+4

ANSWERS 1 a ( 1

3,3127 ) , (−5,77 )

b (3,0 ) , (−3,0 )c (1,5 ) , (1,1 ) , (1−3√2 ,−2 ), (1+3√2 ,−2 )d (0,0 );maximum, (4 ,−9.071 )minimume (−4,0 ); minimum, (4,0 );minimum , (0 ,6.352 );maximum2 No relative extremum at x=0. (1,-1) is a minimum point.3 a f ' ( x )= 1

3 ( x−3 )23

, f ' ' ( x )= 2

9 ( x−3 )53b (3,0)c No extremum point. Inflection point: (1,0)4 a (0,0 ) , (2 ,−16 ) , (−2 ,−16 )b Increasing:(−2,0 )∪(2,+∞) ; Decreasing:(−∞ ,−2 )∪(0,2)c Relative maximum:(0,0) , Relative minimum:(2,-16) and (−2 ,−16 )d Concave up: (−∞,− 2

√3 )∪( 2

√3,+∞) , Concave down: (−2

√3,

2

√3 )e (± 2

√3,−896

81 )5 a

b

c

d

1) Find the intervals where the function is increasing and decreasing.i) [Ans: increasing on ]

ii) [Ans: increasing , decreasing]iii) [Ans: increasing , decreasing]2) Find the critical points for the following functions.

i) [Ans: and ]ii) [Ans: and ]iii) [Ans:

]3) Determine where the function is concave upward and concave downward.

i) [Ans: concave up , concave down]ii) [Ans: concave up]4) Sketch the graph ofi)

Ans:

ii) Ans:

iii) Ans:

iv) Ans:

v) Ans:

vi) Ans:

5) Determine the relative extrema of the function . [Ans: relative maximum: , relative minimum:]

6) Sketch the graph of a function having the following properties:

Ans:

Question

1. Identify critical points and find the maximum and minimum value on the given interval I.a) f(x) = x2 + 2x; I =[ 3

2,12 ]

b) r(θ) = 2 cosθ; I = [−π4 ,π3 ]

c) g(t) =[t 23 ] ; I = [-1, 8]

2. Sketch the grapha) f ( x )=x2 ( x−1 )2 (x+1 )2

b) f ( x )=2x3−3 x2+12x+50

3. For each of the given function;

(0,2)

(-1,4)

(1,0)

f(x)

x

i) find the x and y intercepts (if any).ii) all the asymptotes (if any)iii) the interval of increase and decreaseiv) local maximum / local minimumv) interval of concavityvi) inflection point (if any)vii) sketch the function completely.

(a) f ( x )= 1+√ x1−√ x

(b) f ( x )=4 x−3 x43

(c) f ( x )= 1+2 x

(1−x )2

answers: 1a) Critical points: - 32

, - 1, 12

; maximum value 54

; minimum value - 1

1b) Critical points: - π4

, 0,π3

; maximum value 2; minimum value 1

1c) Critical points: -1, 0, 8; maximum value 4; minimum value 0

Answer 2a

Answer 2b

Answer 3a

Answer 3bAnswer 3c

Question

1) Find the critical numbers and the relative extrema for the functions, if any:

a) y=4 x3+2x2

b) y=x3−3x+3

c) y= (x−3 )25

d) y=|x2−1|e) y=x √8−x2

f) y= x2−3x−2

2) The graph of f ' on (1, 6) is shown below. Find the intervals on which f is increasing or decreasing.

3) The graph of f ', the derivative of a function f, is shown below. Find the relative extrema of f.

4) The graph of f is shown below and f is twice differentiable. Which of the following statements is true:

A. f(5) < f '(5) < f ''(5)B. f ''(5) < f '(5) < f (5)C. f '(5) < f (5) < f ''(5)D. f '(5) < f ''(5) < f (5)E. f ''(5) < f (5) < f '(5)

5)

INTERVAL SIGN OF f '( x) SIGN OF f ' ' (x )x<1 −¿ +¿

1<x<2 +¿ +¿2<x<3 +¿ −¿3<x<4 −¿ −¿

4<x −¿ +¿

A sign chart is presented for the first and second derivative of a function f . Assuming that f is continuous everywhere . Find

a) the interval on which f is increasing and decreasingb) the interval on which f is concave up and down.c) The x-coordinates of all inflection points

6) Find the absolute maximum and minimum values of f on the closed interval, and state where the values occur.

a) f ( x )=4 x2−12 x+10 ; [ 1,2 ]

b) f ( x )=(x−2)3; [1,4 ]c ¿ f ( x )= 3 x

√4 x2+1; [−1,1 ]

d) f ( x )=x−2 sinx ; [−π4 ,π2 ]

e) f ( x )=1−|9−x2|; [ 1,2 ]

7) For each of the given function;

i) find the x and y intercepts (if any).

ii) all the asymptotes (if any)

iii) the interval of increase and decrease

iv) local maximum / local minimum

v) interval of concavity

vi) inflection point (if any)

vii) sketch the function completely.

a¿ f (x )=x4−3 x3+3x2+1b¿ f (x )=3 x2−8x2−4

c ¿ f ( x )= 2 x−x2

x2+x−2

d ¿ f ( x )=(x−2)3

x2

e ¿ f ( x )=x23( 5

2−x )

f ¿ f ( x )=x√4−x2

g¿ f ( x )=x2−1x

8) A curve has the equation y=x3+a x2+bx+c. The curve cuts the y-axis at

y=−13and has stationary points at x=−1and x=−73

.

(a) Find the values of a, b and c.

(b) Find the inflection points

(c) Sketch the graph of f

9) Let f ( x )=x2+ px+q . Find the values of p and q such that f (1 )=3 is an extreme value

of f on [ 0,2 ]. Is this value a maximum or minimum?

10) Find the values of of a ,b , c and d so that the function

f ( x )=a x3+b x2+cx+d

has relative minimum at (0,0 ) and relative maximum at (1,1 ).

Answer

a) 0 ,−13

b) rel max (-1,5) and min (1,1)c) 3d) rel. max (0,1)e) rel max (2,4) and min (-2,-4)f) rel. max (1,2) and min (3,6)

2. decreasing [ 1,2 ]∪ [ 5,6 ], increasing [ 2,5 ]3. rel. max x=-2, rel.min x=34.C5. increasing [ 1.3 ], decreasing (−∞ ,2 ]∪¿ Concave up (−∞ ,2¿∪(4 ,+∞), concave down (2,4)6.

a) max =2 at x=1,2, min =1 at x=3/2b) max =8 at x=4, min=-1 at x =1c) max = 3/√5 at x=1, min −3/√5 at x=-1

d) max =√2−π4

at x=−π4

, min −√3+ π3at x= π

3

7.a)

b)

c)

d)

e)

f)

g)

8.

a) a=5 , b = 7, c = -13b) x=-5/3

9. p = -2, q= 4 , x = 1 is minimum value.

10. a = -2 , b = 3 , c= 0, d = 0

Question

1.f ( x )= x2

x2+4

2.f ( x )= 1

2( x−2 )+ 1

2( x+2)

3 . f ( x )=3 ( x+1 )2

( x−1 )2

4 . f ( x )=4 x3−9 x4

5 . f ( x )=x4−6 x2+5

Task; For each of the given function;i) find the x and y intercepts (if any).ii) all the asymptotes (if any)iii) the interval of inc. and dec.iv) local maximum / local minimumv) interval of concavityvi) inflection point (if any)vii) sketch the function completely.

Answer

1. f ( x )= x2

x2+4 3.f ( x )=

3 ( x+1 )2

(x−1 )2

2.

f ( x )= 12( x−2)

+ 12(x+2 )

5.f ( x )=x4−6 x2+5

f ( x )=4 x3−9x 4

1. Find the critical points for the following functions.(a) f ( x )=x 4−8 x2+3

(b) f ( x )=x32 +4

(c) f ( x )=√x2−64

Ans: (a) (0, 3), (2, -13) and (-2, -13) (b) (0, 4)

(c) (-8, 0), (8, 0)

2. Find the interval where f ( x )=23x3−13 x2

2+6 x+1

is increasing or decreasing.Ans: (−∞, 12 ) , (6 ,∞ ) increasing , ( 1

2,6)decreasing

3. It is given that y=6(5−x )13 .

(a) Find (i) dydx (ii) d

2 ydx2

(b) Find t he coordinates of the critical point, and determine the nature of the point.Ans: (a)(i) − 2

(5−x)23

(ii) −4

3(5−x )53

(b) (5,0) is a point of inflection.4. Sketch the graph for

(a)f ( x )=x3+3 x2−4 (b) f ( x )=−x3−x2

(c)f ( x )=2x4−8 x (d) f ( x )=2x4−8 x2+6

5. Sketch the curve y=9 x−6x+7

.6. Sketch the curve y=1−2

x

7. Sketch the curve y= 4 x

1+x+x2

8. Given f ( x )=3 x4−16 x3+18 x2, with domain (−∞ ,+∞ )

f ' ( x )=12 x3−48 x2+36 x and f ' ' ( x )=36 x2−96 x+36.

i. Using Second Derivative Test, find the relative maximum and/or relative minimum, if any.

ii. Determine the intervals where the function are increasing and decreasing, if any.

9. Given f ( x )=x 4−4 x3+10, with domain (−∞ ,0 )∪ (0 ,+∞ ).

f ' ( x )=4 x3−12x2 and f ' ' ( x )=12x2−24 x.

i. Find the intervals where the function is concaving upwards and downwards.

ii. Find the inflection point(s).

10. Given the following information on the function f (x), hence sketch the graph of the function.

i) Domain is (−∞ ,2 )∪ (2 ,+∞ ).ii) Interval where functions is increasing are (2,5 )iii) Interval where functions is decreasing are (−∞ ,2 )∪(5 ,+∞)iv) Interval where functions is concave upwards are (−5,2 )∪ (4 ,+∞ )v) Interval where functions is concave downwards are (−∞ ,−5 )∪ (2,4 )vi) f (−5 )=2 , f ( 4 )=4 and f (5 )=6vii)

Answer

4alim

x→2−¿ f ( x )=−3 , limx→ 2+ ¿f ( x )=1, lim

x→−∞f ( x )=8,∧ lim

x→+∞f (x ) =0 ¿

¿ ¿¿

4b

4c

4d

5

Horizontal asymptote at y = 9, Vertical asymptote at x = -7

6

7

Minumum point =(-1, -4), Maximum point = (1, 4/3)8 Max pt: (1,5). Min pt : (0, 0), (3, -27)Increase: (−∞ ,0 ]∪ [1,3]Decrease: [ 0,1 ]∪¿9 Concave up: (−∞ ,0 )∪ (2 ,∞)Concave down: (0,2)Inflection point at (0,10) and (2,-6)