A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 12: Differentiation · Lesson 4

Increasing and decreasing functions, and second derivatives

73 marks

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Date

Do Now

Quick questions to start.

A1
Needed today
Solve the inequality $x^{2} - 2 x - 15 < 0$ .
[2 marks]
Answer
A2
Needed today
Write $x^{2} + 6 x + 14$ in the form $(x + a)^{2} + b$ .
[2 marks]
Answer
A3
Last lesson
Find an equation of the tangent to the curve $y = x^{2} - 5 x + 1$ at the point where $x = 4$ .
[2 marks]
Answer $y = \dots\dots$
A4
From chapter 2
The equation $x^{2} + k x + 9 = 0$ has no real roots. Find the set of possible values of $k$ .
[2 marks]
Answer

A-Level

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

Example 1

Find the interval on which the function $f (x) = 2 x^{3} - 9 x^{2} - 24 x + 5$ is decreasing.

Hint

f

is decreasing where

f^{'} (x) < 0

. Factorise

f^{'} (x)

and solve the quadratic inequality.

B1
Find the set of values of $x$ for which the function $f (x) = x^{3} + 6 x^{2} - 36 x + 2$ is increasing.
[4 marks]
Answer

Example 2

Show that the function $f (x) = x^{3} - 3 x^{2} + 5 x - 2$ is increasing for all real values of $x$ .

Hint Differentiate, then complete the square on

f^{'} (x)

to show it is always positive.

B2
Show that the function $f (x) = 1 - 4 x + 3 x^{2} - x^{3}$ is decreasing for all real values of $x$ .
[4 marks]

Example 3

Given that $y = \frac{2 x ^{4} - 3}{x}$ , find $\frac{d y}{d x}$ and $\frac{d ^{2} y}{d x ^{2}}$ .

Hint Split the fraction into powers of

x

first. Then differentiate twice.

B3
Given that $f (x) = 8 x + \frac{2}{x}$ , $x > 0$ , find $f^{''} (x)$ . Hence find the value of $f^{''} (4)$ .
[4 marks]
Answer $f^{''} (4) = \dots\dots$

A-Level

C1
Find the set of values of $x$ for which $f (x) = x - 6 x$ , $x > 0$ , is decreasing.
[4 marks]
Answer

C2
Given that $y = (x^{2} - 3) (2 x + 1)$ , find the value of $x$ for which $\frac{d ^{2} y}{d x ^{2}} = 0$ .
[3 marks]
Answer $x = \dots\dots$
C3
$f (x) = a x^{3} + b x^{2} - 5 x$ , where $a$ and $b$ are constants. Given that $f^{'} (2) = 7$ and $f^{''} (2) = 18$ , find the value of $a$ and the value of $b$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$
C4
The function $f (x) = x^{3} - p x$ , where $p$ is a constant, is decreasing on the interval $[- 2, 2]$ . Find the set of possible values of $p$ .
[3 marks]
Answer $p \dots\dots$

Edexcel exam style

Stretch

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

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find where a function is increasing or decreasing.
I can show a function is increasing or decreasing for all x.
I can find and use a second derivative.
I can use derivatives for velocity and acceleration.

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