A-Level Maths (Edexcel 9MA0) · Pure Year 2 · Chapter 9: Differentiation · Lesson 7

Concave and convex functions

111 marks

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Quick questions to start.

A1
Needed today
Given $f (x) = x^{4} - 4 x^{3}$ , find $f^{''} (x)$ .
[2 marks]
Answer
A2
Needed today
Given $f (x) = x e^{2 x}$ , find $f^{''} (x)$ . Give it in the form $e^{2 x} (a + b x)$ .
[2 marks]
Answer
A3
Needed today
Solve the inequality $x^{2} - 4 x + 3 < 0$ .
[2 marks]
Answer
A4
Common slip
For $y = x^{4}$ , $\frac{d ^{2} y}{d x ^{2}} = 0$ when $x = 0$ . Is the origin a point of inflection? Give a reason.
[2 marks]
Answer
A5
From chapter 9
Find the coordinates of the stationary point on $y = x^{2} - 6 x + 1$ and say whether it is a maximum or a minimum.
[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 interval on which $f (x) = x^{3} - 6 x^{2} + 5 x$ is concave, and the interval on which it is convex.

Hint Concave where

f^{''} (x) ⩽ 0

, convex where

f^{''} (x) ⩾ 0

B1
Find the set of values of $x$ for which $f (x) = x^{4} - 6 x^{2} + 3 x$ is concave, and the set for which it is convex.
[4 marks]
Answer

Example 2

Show that the curve $y = x^{3} - 3 x^{2} + 4$ has one point of inflection, and find its coordinates.

Hint Solve

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

, then check that the sign changes on either side.

B2
Find the coordinates of the point of inflection on the curve $y = 2 x^{3} + 3 x^{2} - 12 x + 1$ . Show that the second derivative changes sign there.
[4 marks]
Answer

Example 3

Show that $f (x) = x ln x$ is convex for all $x > 0$ .

Hint Convex for all

x

means

f^{''} (x) ⩾ 0

for every

x

in the domain.

B3
Show that $f (x) = ln x - x^{2}$ is concave for all $x > 0$ .
[4 marks]

Example 4

Find the coordinates of the point of inflection on $y = x e^{- x}$ .

Hint Use the product rule twice and keep

e^{- x}

as a factor.

e^{- x}

is never zero.

B4
Find the exact coordinates of the point of inflection on $y = (x + 1) e^{- x}$ .
[5 marks]
Answer

Example 5

Find and classify the stationary points of $y = x^{4} - 4 x^{3}$ , and find any other point of inflection.

Hint At a stationary point where

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

, look at the sign of

\frac{d y}{d x}

either side.

B5
Find and classify the stationary points of $y = x^{5} - 5 x^{4}$ , and find the coordinates of the point of inflection.
[6 marks]
Answer

Practice

A-Level

C1
Asha says: "For $y = x^{4} - 4 x$ , $\frac{d ^{2} y}{d x ^{2}} = 0$ at $x = 0$ , so $(0, 0)$ is a point of inflection." Explain why Asha is wrong, and say whether the curve is concave or convex.
[4 marks]

C2
$f (x) = x^{3} + 3 x^{2} - 9 x$ . Find the set of values of $x$ for which $f$ is convex, and the coordinates of the point of inflection.
[4 marks]
Answer
C3
Find the coordinates of the points of inflection on $y = sin x + cos x$ for $0 ⩽ x ⩽ 2 π$ .
[5 marks]
Answer
C4
Ben differentiates $y = x^{2} ln x$ and writes $\frac{d y}{d x} = 2 x \times \frac{1}{x} = 2$ , so $\frac{d ^{2} y}{d x ^{2}} = 0$ and "the curve has no points of inflection." Explain his mistake, and find the exact coordinates of the point of inflection.
[5 marks]
Answer

Exam-style questions

Edexcel exam style

D1
A curve has equation $y = x^{3} - 6 x^{2} + 9 x + 2$ .
(a)
Find the coordinates of the stationary points, and determine their nature.
[4 marks]
Answer
(b)
Find the coordinates of the point of inflection.
[2 marks]
Answer
(c)
Find the set of values of $x$ for which the curve is both concave and increasing.
[3 marks]
Answer
Total for D1: 9 marks

D2
Sam is studying $f (x) = 3 x^{5} - 10 x^{4}$ . He writes: " $f^{''} (x) = 0$ at $x = 0$ and at $x = 2$ , so the curve has two points of inflection."
Sam is wrong.
(a)
Show that $f^{''} (x) = 60 x^{2} (x - 2)$ .
[2 marks]
(b)
Explain why the origin is not a point of inflection.
[3 marks]
(c)
Find the coordinates of the point of inflection, and the set of values of $x$ for which $f$ is convex.
[4 marks]
Answer
Total for D2: 9 marks
D3
The curve $y = x^{4} + a x^{3} + 6 x^{2}$ has no points of inflection. Find the set of possible values of the constant $a$ .
[7 marks]
Answer
D4
A curve has equation $y = e^{- x^{2}}$ .
(a)
Show that $\frac{d ^{2} y}{d x ^{2}} = (4 x^{2} - 2) e^{- x^{2}}$ .
[3 marks]
(b)
Find the exact coordinates of the points of inflection.
[3 marks]
Answer
(c)
State the set of values of $x$ for which the curve is concave.
[2 marks]
Answer
Total for D4: 8 marks
D5
The diagram shows the curve $y = x + 2 cos x$ for $0 ⩽ x ⩽ 2 π$ .
(a)
Find the exact coordinates of the stationary points, and determine their nature.
[5 marks]
Answer
(b)
Find the coordinates of the points of inflection.
[2 marks]
Answer
(c)
Find the set of values of $x$ in $0 ⩽ x ⩽ 2 π$ for which the curve is concave.
[3 marks]
Answer
Total for D5: 10 marks
D6
$f (x) = \frac{ln x}{x}$ , $x > 0$ .
(a)
Show that $f^{''} (x) = \frac{2 ln x - 3}{x ^{3}}$ .
[4 marks]
(b)
Find the exact coordinates of the point of inflection, and the set of values of $x$ for which $f$ is convex.
[4 marks]
Answer
Total for D6: 8 marks

Extension

Stretch

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

E1
The curve $y = x^{3} + a x^{2} + b x + c$ has a point of inflection at $(1, - 2)$ , and the gradient of the curve there is $- 3$ . Find $a$ , $b$ and $c$ . Then show that the tangent to the curve at $(1, - 2)$ meets the curve at no other point.
[9 marks]
Hint 1 · What to try
Write down $\frac{d y}{d x}$ and $\frac{d ^{2} y}{d x ^{2}}$ . The point of inflection gives one equation, the gradient a second, and the point on the curve a third.
Hint 2 · The first line
$\frac{d ^{2} y}{d x ^{2}} = 6 x + 2 a$ is $0$ at $x = 1$ , so $a = - 3$ .
Hint 3 · The full method
Set the curve equal to the tangent and factorise. You already know one root: it is a repeated one.
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can find the interval on which a function is concave or convex using the second derivative.
I can find a point of inflection and check that the second derivative changes sign there.
I can explain why a second derivative of zero is not enough for a point of inflection.
I can classify stationary points, including a stationary point of inflection.

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