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3.02 The second derivative and properties of graphs

Worksheet
Nature of stationary points
1

For each of the following functions:

i
Find the x-coordinate of the turning point.
ii
Find the value of the second derivative at this point.
iii
Hence, classify the stationary point.
a
f \left( x \right) = \left(x - 5\right)^{2} + 3
b
f \left( x \right) = - \left(x - 5\right)^{2} - 4
c
f \left( x \right) = \left(x + 4\right) \left(x + 2\right)
d
f \left( x \right) = - \left(x + 4\right) \left(x + 6\right)
2

Consider the function f \left( x \right) = 3 \left(x - 4\right)^{3} + 1.

a

By inspecting the equation and/or the graph of the function, state what type of stationary point the function has.

b

Use the first derivative to find the x-coordinate of the stationary point.

c

Find the second derivative at this point.

d

Hence confirm what type of stationary point the function has.

3

For each of the following functions:

i

Find the x-coordinates of the stationary points.

ii

Determine the nature of the stationary points.

a
f \left( x \right) = 4 x^{3} + 8 x^{2} + 5 x + 6
b
f \left( x \right) = 3 \left(x - 7\right)^{2} \left(x + 2\right)
c
f \left( x \right) = \dfrac{8}{x^{2}} + 2 x
d
f \left( x \right) = - \dfrac{4}{x^{2}} + 8 x
4

For each of the following functions:

i

Find the derivative.

ii

Find the stationary point(s).

iii

Classify each stationary point.

a
y = - 6 x^{2} + 84 x - 29
b
y = x^{3} - 21 x^{2} + 144 x - 19
5

Consider the function f \left( x \right) = 2 x^{3} - 18 x^{2} + 54 x - 49.

a

Find the x-coordinate(s) of the stationary point(s).

b

Find the x-coordinate(s) of the point(s) of inflection.

c

Determine the nature of the stationary point(s).

6

Consider the function f \left( x \right) = 4 x^{3} + 8 x^{2} + 5 x + 2.

a

Find the x-coordinate(s) of the stationary point(s).

b

Find the x-coordinate(s) of the point(s) of inflection.

7

Consider the function y = 5 - \left(x - 4\right)^{4}.

a

Find the coordinates of the stationary point.

b

Explain why the point is a turning point and not a point of inflection.

8

Consider the function y = \left(x^{2} - 5\right)^{3}.

a

Find y''.

b

Find the points of inflection.

c

Which of these points are horizontal points of inflection?

9

Consider the function f \left( x \right) = - 4 \sin \left(x + \dfrac{\pi}{6}\right) on the interval 0 \leq x \leq 2 \pi.

a

Find the coordinates of any turning points.

b

Find an expression for f'' \left( x \right).

c

Hence classify the turning points.

Features of graphs
10

For each of the following quadratic functions:

i

State whether the leading coefficient of f \left( x \right) is positive or negative.

ii

Hence, determine the nature of the turning point.

iii

Find f' \left( x \right).

iv

Find f'' \left( x \right).

v

State whether the curve is concave up or down.

a

f \left( x \right) = x^{2} - 4 x + 9

b

f \left( x \right) = - x^{2} + 4 x - 9

11

Consider the function y = \left(x + 4\right)^{2} \left(x + 1\right) graphed below:

a

State the x-coordinates of the turning points of the function.

b

What is the gradient at these turning points?

c

State the x-coordinate of the point of inflection.

d

What sign is the gradient of the function at the point of inflection?

e

Sketch the graph of a possible gradient function.

-6
-5
-4
-3
-2
-1
1
2
x
-4
-3
-2
-1
1
2
3
4
y
12

Consider the function y = - \left(x + 13\right) \left(x - 2\right) \left(x - 11\right) graphed below.

-13
-12
-11
-10
-9
-8
-7
-6
-5
-4
-3
-2
-1
1
2
3
4
5
6
7
8
9
10
11
12
13
x
-1000
-900
-800
-700
-600
-500
-400
-300
-200
-100
100
200
300
400
500
y
a

State the x-coordinates of the turning points of the function.

b

State the x-coordinate of the point of inflection.

c

What sign is the gradient at the point of inflection?

13

Consider the function f \left( x \right) = \left( 3 x + 8\right) \left(x + 1\right).

a

Determine an equation for the gradient function f' \left( x \right).

b

State the coordinates of the turning point.

c

Determine the nature of the turning point.

d

Find the absolute minimum value of the function.

14

Consider the function f \left( x \right) = x^{3} - 12 x - 2.

a

Determine an equation for the gradient function f' \left( x \right).

b

State the coordinates of the turning points.

c

Determine an equation for the second derivative f'' \left( x \right).

d

Determine which turning point is a local minimum and which turning point is a local maximum.

e

Is - 18 the absolute minimum value of the function? Explain your answer.

15

Consider the function f \left( x \right) = 4 x^{3} + 5 x^{2} - 4.

a

Find an equation for the gradient function f' \left( x \right).

b

Find the coordinates of the stationary points.

c

Determine which turning point is a local minimum and which turning point is a local maximum.

d

Is - 4 the absolute minimum value of the function? Explain your answer.

16

Consider the function f \left( x \right) = 10 x \sqrt{x + 3}.

a

Determine an equation for the gradient function f' \left( x \right).

b

Find the coordinates of the turning point.

c

Determine whether the point \left( - 2 , - 20 \right) is a minimum or maximum turning point.

d

Is - 20 the absolute minimum?

17

Consider the function f \left( x \right) = -3 x^{2} + 12 x + 2.

a

Find f' \left( x \right).

b

Find f'' \left( x \right).

c

Describe the rate of change of the gradient.

d

Are there any points of inflection on f(x)? Explain your answer.

18

Consider the function y = \left(x + 6\right)^{3}.

a

State the transformation that turns y = x^{3} into y = \left(x + 6\right)^{3}.

b

Find the point of inflection of y = x^{3}.

c

Find the point of inflection of y = \left(x + 6\right)^{3}.

d

Complete the following table of values:

x- 7- 6- 5
y'
y''0
e

Is the point of inflection a horizontal point of inflection? Explain your answer.

f

For what values of x is the graph concave up?

19

Consider the function y = 4 x^{3} - 16 x^{2} + 4 x + 6.

a

Find y''.

b

Find the point of inflection.

c

Is the point of inflection a horizontal or ordinary point of inflection?

d

For what values of x is the graph concave down?

20

Consider the function y = x^{4} - 8 x^{3} - 9.

a

Find y''.

b

Find the points of inflection.

c

Complete the following table of values:

x- 20246
y'
y''00
d

Classify each point of inflection as an ordinary or horizontal point of inflection.

e

For what values of x is the graph concave up?

21

Consider the functiony = x e^{x}.

a

Find y'.

b

Find the x-coordinate of any stationary points.

c

Find the x-coordinate of any possible points of inflection.

d

Complete the table of value to confirm that there exists a point of inflection:

x - 3 - 2 - 1
y''
e

For what values of x is the graph of y = x e^{x} concave up?

f

What type of stationary point is at x = -1?

22

Consider the function y = x^{5} - 3 x^{2}.

a

Find y''.

b

Find the x-coordinate of the potential point of inflection.

c

Is the point of inflection a horizontal or ordinary point of inflection?

d

For what values of x is the graph concave up?

e

For what values of x is the graph concave down?

23

Consider the function f \left( x \right) = \left(x - 8\right) \left(x - 5\right)^{2}.

a

Find f' \left( x \right).

b

Find the turning points.

c

Find f'' \left( x \right).

d

Classify the turning points.

e

For what values of x is the graph concave down?

24

Consider the function y = x \left(x - 3\right)^{2}.

a

Find the coordinates of the turning points.

b

Classify both stationary points.

c

Find the coordinates of the possible point of inflection.

d

Complete the table of values to prove that this is a point of inflection:

x123
y\rq\rq0
25

The first derivative of a certain function is f' \left( x \right) = 3 x^{2} + 9 x.

a

Determine the interval over which the function is increasing.

b

Determine the interval over which the function is decreasing.

c

Find f'' \left( x \right).

d

Determine the interval over which the function is concave up.

e

Determine the interval over which the function is concave down.

f

Find the x-coordinate of the maximum turning point.

g

Find the x-coordinate of the potential point of inflection.

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Outcomes

4.1.1.1

understand the concept of the second derivative as the rate of change of the first derivative function

4.1.1.3

understand the concepts of concavity and points of inflection and their relationship with the second derivative

4.1.1.4

understand and use the second derivative test for finding local maxima and minima

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