AQA Level 2 Further Maths · Level 2 Certificate · Chapter 8: Calculus · Lesson 3

Increasing and decreasing functions, and the second derivative

About 75 minutesNo calculator71 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/further-maths-increasing-and-decreasing-functions-and-the-second-derivative/

Date

Do Now

The first questions are what today's lesson needs. The others bring back earlier topics.

A1
Needed today
Solve $x^{2} - 3 x - 18 < 0$ .
[2 marks]
Answer $\dots\dots < x < \dots\dots$
A2
Needed today
Write $3 x^{2} - 12 x + 20$ in the form $a (x + b)^{2} + c$ .
[2 marks]
Answer
A3
Last lesson
Work out the gradient of the normal to the curve $y = x^{3} - 2 x$ at the point where $x = 2$ .
[2 marks]
Answer
A4
From chapter 4
Simplify $\frac{( 2 x ^{3} ) ^{2}}{4 x}$ .
[2 marks]
Answer

Example, then your turn

Level 2

Your teacher works through each example with you. Then try the one beside it.

Example 1

Work out the values of $x$ for which $y = x^{2} - 6 x + 1$ is an increasing function.

Hint A function is increasing where

\frac{d y}{d x} > 0

B1
Work out the values of $x$ for which $y = 7 + 10 x - x^{2}$ is a decreasing function.
[3 marks]
Answer $x \dots\dots$

Example 2

Work out the values of $x$ for which $y = x^{3} - 3 x^{2} - 24 x + 5$ is a decreasing function.

Not drawn accurately

Hint Solve

\frac{d y}{d x} < 0

. Find the critical values first, then decide which side of them the gradient is negative.

B2
Work out the values of $x$ for which $y = 2 x^{3} + 3 x^{2} - 36 x$ is an increasing function.
[4 marks]
Answer $\dots\dots\dots\dots$

Example 3

Prove that $y = x^{3} + 3 x^{2} + 5 x - 2$ is an increasing function for all values of $x$ .

Hint Show that

\frac{d y}{d x} > 0

for every

x

. Complete the square on

\frac{d y}{d x}

B3
Prove that $y = 1 - 5 x + 3 x^{2} - x^{3}$ is a decreasing function for all values of $x$ .
[4 marks]

Example 4

Work out the values of $x$ for which $y = x + \frac{9}{x}$ is an increasing function.

Not drawn accurately

Hint Write

y = x + 9 x^{- 1}

. The function is not defined at

x = 0

B4
Work out the values of $x$ for which $y = 4 x + \frac{25}{x}$ is a decreasing function.
[4 marks]
Answer $\dots\dots\dots\dots$

Example 5

$y = (2 x - 1) (x^{2} + 3)$ . Work out $\frac{d ^{2} y}{d x ^{2}}$ . Hence work out the rate of change of the gradient of the curve at the point where $x = 2$ .

Hint Multiply out first.

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

is the derivative of

\frac{d y}{d x}

: it is the rate of change of the gradient.

B5
$y = (x^{2} - 4) (3 x + 1)$ . Work out $\frac{d ^{2} y}{d x ^{2}}$ . Hence work out the rate of change of the gradient of the curve at the point where $x = - 1$ .
[4 marks]
Answer $\frac{d ^{2} y}{d x ^{2}} = \dots\dots\dots$ and $\dots\dots$

Practice

Level 2

C1
Work out the values of $x$ for which $y = 2 x^{2} + 9 x - 5$ is a decreasing function.
[2 marks]
Answer $x \dots\dots$
C2
Prove that $y = \frac{1}{3} x^{3} - 3 x^{2} + 11 x$ is an increasing function for all values of $x$ .
[3 marks]

C3
$y = 2 x^{3} - \frac{8}{x}$ . Work out the value of $\frac{d ^{2} y}{d x ^{2}}$ when $x = 1$ .
[3 marks]
Answer
C4
Work out the values of $x$ for which the gradient of the curve $y = x^{3} - 6 x^{2} + 4$ is increasing.
[3 marks]
Answer $x \dots\dots$
C5
$y = x^{4} - 6 x^{2} + 2$ . Work out the values of $x$ for which $\frac{d ^{2} y}{d x ^{2}} = 0$ .
[3 marks]
Answer $x = \dots\dots$ and $x = \dots\dots$

Exam-style questions

AQA exam style

D1
Kai is working out the values of $x$ for which $y = x^{3} - 27 x$ is a decreasing function. He writes: $3 x^{2} - 27 < 0$ , so $x^{2} < 9$ , so $x < 3$ .
Explain Kai's mistake. Work out the correct values of $x$ .
[3 marks]
Answer $\dots\dots < x < \dots\dots$

D2
$f (x) = 2 x^{3} - 6 x^{2} + 9 x + 1$ . Use differentiation to show that $f (x)$ is an increasing function for all values of $x$ .
[4 marks]
D3
Work out the values of $x$ for which $f (x) = \frac{2}{3} x^{3} - \frac{1}{2} x^{2} - 6 x$ is a decreasing function. Give your answer as an inequality.
[5 marks]
Answer $\dots\dots < x < \dots\dots$
D4
A ball is thrown upwards. Its height, $h$ metres, above the ground $t$ seconds after it is thrown is $h = 1.5 + 14 t - 5 t^{2}$ .
(a)
Work out $\frac{d h}{d t}$ .
[1 mark]
Answer $\frac{d h}{d t} = \dots\dots\dots$
(b)
Work out the values of $t$ for which the height of the ball is increasing.
[2 marks]
Answer $\dots\dots\dots\dots$
(c)
Work out $\frac{d ^{2} h}{d t ^{2}}$ . Say what it tells you about the ball.
[2 marks]
Total for D4: 5 marks
D5
$y = f (x)$ is the graph of a cubic function. $f (x) > 0$ for $x < 4$ and $f (x) \leq 0$ for $x \geq 4$ . The function is decreasing for $x < - 3$ , increasing for $- 3 < x < 1$ and decreasing for $x > 1$ . Draw a possible sketch of $y = f (x)$ for values of $x$ from $- 5$ to $5$ on the axes.
[4 marks]
D6
$y = x^{3} - 6 x^{2} + k x + 2$ , where $k$ is an integer. $\frac{d y}{d x} > 0$ for all values of $x$ . Work out the least possible value of $k$ .
[4 marks]
Answer $k = \dots\dots$

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
Work out the values of $x$ for which $y = x^{2} + \frac{16}{x}$ is a decreasing function.
[5 marks]
Hint 1 · What to try
Work out $\frac{d y}{d x}$ and solve $\frac{d y}{d x} < 0$ . Be careful when you multiply an inequality by something that might be negative.
Hint 2 · The first line
$\frac{d y}{d x} = 2 x - \frac{16}{x ^{2}}$ . Multiply by $x^{2}$ , which is positive for every $x \neq = 0$ .
Hint 3 · The full method
$2 x^{3} - 16 < 0$ , so $x^{3} < 8$ and $x < 2$ . The function is not defined at $x = 0$ , so it is decreasing for $x < 0$ and for $0 < x < 2$ .
Answer $\dots\dots\dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out where a function is increasing or decreasing.
I can prove that a function is increasing, or decreasing, for all values of $x$ .
I can work out the second derivative and the rate of change of the gradient.
I can say what $\frac{d h}{d t}$ and $\frac{d ^{2} h}{d t ^{2}}$ mean in a context.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers