Topics
11. Differentiation
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iGCSE (2021 Edition)

11.09 Using the second derivative

Lesson
Worksheet
Practice
Worksheet
Higher derivatives
1

Suppose y = \dfrac{2}{x^{5}}. Find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

c

\dfrac{d^{3} y}{d x^{3}}

2

Suppose y = 5 x^{8}. Find:

a

\dfrac{d y}{d x}

b

\dfrac{d^{2} y}{d x^{2}}

c

\dfrac{d^{3} y}{d x^{3}}

d

\dfrac{d^{4} y}{d x^{4}}

e

\dfrac{d^{5} y}{d x^{5}}

3

Consider the function y = - 2 x^{4} + 4 x^{3} - 7 x^{2} + 8 x - 9. Find:

a

\dfrac{d y}{d x}

b

The value of \dfrac{d y}{d x} when x = 3

c

\dfrac{d^{2} y}{d x^{2}}

d

The value of \dfrac{d^{2} y}{d x^{2}} when x = 4

e

\dfrac{d^{3} y}{d x^{3}}

f

The value of \dfrac{d^{3} y}{d x^{3}} when x = 6

4

Find the second derivative of the following functions:

a
y = x^4 - 6x^3 + 8x^2 - 10
b
f \left( x \right) = x^{\frac{1}{7}}
c
f \left( x \right) = \left( 3 x + 2\right)^{4}
d
f \left( x \right) = \dfrac{4}{\left(x + 8\right)^{2}}
e
y = \left(7 - 6 x^{2}\right)^{\frac{1}{3}}
f
y = 4 \sqrt{6 - x}
g
f \left( x \right) = x^{2} \left(x^{3} + 3\right)
h
f \left( x \right) = \left(x^{4} + 2\right) \left(3 - 2 x^{3}\right)
Concavity
5

Consider the given function y = f(x) with points A, B, C and D:

a

At which of these points is f''(x) > 0 ?

b

At which of these points is f''(x) <0 ?

x
y
6

For each of the following curves, state the sign of f'(a) and f''(a):

a
x
y
b
x
y
c
x
y
d
x
y
7

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

8

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

a

Find the first derivative.

b

Find the second derivative.

c

Complete the following tables of values:

x01234
f' \left( x \right)3
d

Describe the rate of change of the gradient.

e

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

9

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.

10

For each of the following functions:

i

State the values of x for which the graph of the function is concave up.

ii

State the values of x for which the graph of the function is concave down.

iii

Explain why there is no point of inflection.

a

y=\sqrt{x+1}

b

y=-\sqrt{x-2}

c

y=\dfrac{1}{x}

d

y=\dfrac{1}{x^2}

11

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:

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?

x- 7- 6- 5
y'
y''0
12

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

a

State the transformation that turns y = x^{3} into y = \left(x + 5\right)^{3}.

b

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

c

Find the point of inflection of y = \left(x + 5\right)^{3}.

d

Complete the following table of values:

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?

x- 6- 5- 4
y'
y''0
13

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:

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?

x- 20246
y'
y''00
14

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 an ordinary point of inflection?

d

For what values of x is the graph concave down?

15

Consider the function y = 3 x^{3} + 6 x^{2} + 8 x + 3.

a

Find y''.

b

Find the point of inflection.

c

Is the point of inflection, an ordinary or horizontal point of inflection?

d

For what values of x is the graph concave down?

16

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

a

Find y''.

b

Find the exact point of inflection.

c

Is the point of inflection an ordinary or a horizontal 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?

17

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.

Stationary points
18

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

a

Find the x-coordinate of the turning point.

b

Find the value of the second derivative at this point.

c

Hence, classify the stationary point.

19

Consider the function f(x)=(x-5)(x+4)^2.

a

Find the turning points.

b

Find the second derivative.

c

Determine the nature of the turning points.

d
State the values of x where the graph of f(x) is concave down.
20

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

a

Find y'.

b

Find y''.

c

Find the stationary points.

d

Classify the stationary points.

e

Find the points of inflection.

f

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

g

For what values of x is the graph concave up?

21

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?

22

For each of the following functions:

i

Find the turning points.

ii

Classify the turning points.

a

f(x)=x^3-27x-7

b

f(x)=x^3-12x-2

c

f(x)=2x^3+9x^2-24x

d

f(x)=x^4-2x^2+3

23

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) = 7 x^{3} + 8 x^{2} + 3 x + 4
b
f \left( x \right) = 8 \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) = 2 x^{3} - 30 x^{2} + 150 x - 247
e
f \left( x \right) = 2 \left(x - 5\right)^{3} + 1
24

Consider the function f \left( x \right) = 5 x^{3} + 7 x^{2} + 3 x + 6.

a

Find the x-coordinates of stationary points.

b

Find the x-value of the point of inflection.

c

How many stationary points are there?

25

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?

26

For each of the following functions:

i

Find the stationary point.

ii

Find the possible point of inflection.

iii

Determine whether this point is a turning point or a point of inflection. Explain your answer.

a
y = 4 - \left(x - 5\right)^{4}
b
y = 5 - \left(x - 4\right)^{4}
c
y = 6 - \left(x - 8\right)^{4}
27

The function f \left( x \right) = a x^{2} + \dfrac{b}{x^{2}} has turning points at x = 1 and x = - 1.

a

Use the fact that there is a turning point at x = 1 to form an equation for a in terms of b.

b

Use the fact that there is a turning point at x = - 1 to form an equation for a in terms of b.

c

What can you deduce about the values of a and b?

28

The function f \left( x \right) = a x^{3} + b x^{2} + 12 x + 5 has a horizontal point of inflection at x = 1.

a

Use information about f' \left( x \right) to write an equation involving a and b.

b

Use information about f'' \left( x \right) to write an equation for b in terms of a.

c

Hence, find the value of a.

d

Hence, find b.

29

The function f \left( x \right) = a x^{3} + 18 x^{2} + c x + 4 has turning points at x = 4 and x = 2. Find the value of:

a

a

b

c

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Outcomes

0606C14.2

Use the notations f'(x), f''(x), dy/dx, d^2y/dx^2 [=d/dx(dy/dx)].

0606C14.5B

Apply differentiation to stationary points.

0606C14.6

Use the first and second derivative tests to discriminate between maxima and minima.

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