A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 12: Differentiation · Lesson 5

Stationary points and gradient functions

75 marks

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Date

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Quick questions to start.

A1
Needed today
Solve $3 x^{2} - 12 x - 15 = 0$ .
[2 marks]
Answer
A2
Last lesson
Given that $y = 2 x^{3} - 9 x^{2} + 7$ , find $\frac{d ^{2} y}{d x ^{2}}$ .
[2 marks]
Answer $\frac{d ^{2} y}{d x ^{2}} = \dots\dots$
A3
From chapter 7
Factorise fully $x^{3} - 2 x^{2} - 5 x + 6$ .
[2 marks]
Answer
A4
From chapter 2
Find the range of $f (x) = x^{2} - 6 x + 11$ .
[2 marks]
Answer

Example, then your turn

A-Level

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

Example 1

Find the coordinates of the stationary points on the curve $y = 2 x^{3} + 3 x^{2} - 36 x + 5$ , and determine their nature.

Hint Solve

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

. Then put each

x

into

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

: negative means a maximum, positive a minimum.

B1
Find the coordinates of the stationary points on the curve $y = x^{3} - 3 x^{2} - 24 x + 10$ , and determine their nature.
[5 marks]

Example 2

Find the stationary points on the curve $y = x^{4} - 4 x^{3} + 6$ , and determine their nature.

Hint At one point

\frac{d ^{2} y}{d x ^{2}} = 0

, which tells you nothing. Look at the sign of

\frac{d y}{d x}

just either side.

B2
Find the stationary points on the curve $y = 3 x^{4} + 8 x^{3} - 1$ , and determine their nature.
[6 marks]

Example 3

The diagram shows the cubic $y = f (x)$ , with turning points $(- 1, 6)$ and $(2, - 3)$ . For $y = f^{'} (x)$ , find where it meets the $x$ -axis, where $f^{'} (x) < 0$ , and the $x$ -coordinate of its turning point.

Not drawn accurately

Hint A turning point of

y = f (x)

is where

f^{'} (x) = 0

. Where

f

goes down,

f^{'} (x)

is below the

x

-axis.

B3
The diagram shows $y = f (x)$ , with turning points $(- 2, 7)$ and $(2, - 1)$ and the asymptote $y = 3$ . For $y = f^{'} (x)$ , find where it meets the $x$ -axis, its horizontal asymptote, and where $f^{'} (x) > 0$ .
[4 marks]
Not drawn accurately

Practice

A-Level

C1
$f (x) = 5 + 32 x - x^{4}$ . Find the greatest value of $f (x)$ , and hence state the range of $f$ .
[4 marks]
Answergreatest value $\dots\dots$ ; range $\dots\dots$

C2
Find the coordinates of the stationary points on the curve $y = x (x - 6)^{2}$ , and determine their nature.
[5 marks]
C3
The curve $y = x^{3} + k x^{2} + 3 x$ , where $k$ is a constant, has no stationary points. Find the set of possible values of $k$ .
[3 marks]
Answer

Exam-style questions

Edexcel exam style

D1
$f (x) = x^{4} - 2 x^{3} - 3 x^{2} + 4 x + 1$ . You are given that $f^{'} (2) = 0$ .
(a)
Find the coordinates of the three stationary points of the curve $y = f (x)$ .
[4 marks]
(b)
Determine the nature of each stationary point.
[3 marks]
(c)
Find the set of values of $k$ for which the equation $f (x) = k$ has exactly four real solutions.
[2 marks]
Answer
Total for D1: 9 marks

D2
The curve $y = x^{5} - 5 x^{4} + 3$ has a stationary point at $(0, 3)$ . Jo writes: " $\frac{d ^{2} y}{d x ^{2}} = 0$ at $x = 0$ , so $(0, 3)$ is a point of inflection."
Jo's conclusion is wrong.
(a)
Find the coordinates of both stationary points of the curve.
[4 marks]
Answer $(\dots\dots, \dots\dots)$ and $(\dots\dots, \dots\dots)$
(b)
Use $\frac{d ^{2} y}{d x ^{2}}$ to find the nature of the stationary point where $x = 4$ .
[2 marks]
Answer $(4, - 253)$ is a $\dots\dots$
(c)
Show that $(0, 3)$ is a maximum, and so explain why Jo is wrong.
[3 marks]
Total for D2: 9 marks
D3
The curve $C$ has equation $y = x + \frac{32}{x ^{2}}$ , $x > 0$ .
(a)
Find the coordinates of the stationary point of $C$ .
[4 marks]
Answer $(\dots\dots, \dots\dots)$
(b)
Determine the nature of the stationary point.
[2 marks]
Answer $\dots\dots$
(c)
State the range of values of $y$ on $C$ .
[1 mark]
Answer
Total for D3: 7 marks
D4
The curve $y = a x^{3} + b x^{2} - 12 x + 4$ , where $a$ and $b$ are constants, has a stationary point at $(1, - 3)$ .
(a)
Find the value of $a$ and the value of $b$ .
[4 marks]
Answer $a = \dots\dots$ $b = \dots\dots$
(b)
Find the coordinates of the other stationary point.
[3 marks]
Answer $(\dots\dots, \dots\dots)$
(c)
Determine the nature of each stationary point.
[2 marks]
Answer $(1, - 3)$ is a $\dots\dots$ ; $(- 2, 24)$ is a $\dots\dots$
Total for D4: 9 marks

Extension

Stretch

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

E1
The curve $y = x^{3} - 3 k x^{2} + 3 (k^{2} - 1) x$ depends on the constant $k$ . Show that, for every value of $k$ , the curve has a maximum and a minimum whose $x$ -coordinates differ by $2$ . Find the difference between the $y$ -coordinates of the maximum and the minimum.
[6 marks]
Hint 1 · What to try
Factor $3$ out of $\frac{d y}{d x}$ and complete the square in $x$ . Write it in terms of $(x - k)$ .
Hint 2 · The first line
$\frac{d y}{d x} = 3 ((x - k)^{2} - 1)$ , so the stationary points are at $x = k - 1$ and $x = k + 1$ .
Hint 3 · The full method
Put $x = k + u$ . Then $y = u^{3} - 3 u + (k^{3} - 3 k)$ . The stationary points are at $u = \pm 1$ , and the last bracket is the same for both.
Answerdifference $= \dots\dots$

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find stationary points and decide whether each is a maximum or a minimum.
I can spot a point of inflection when the second derivative is zero.
I can use a greatest or least value to find a range.
I can sketch a gradient function from a curve.

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