AQA Level 2 Further Maths · Level 2 Certificate · Chapter 8: Calculus · Lesson 4

Stationary points

About 80 minutesNo calculator89 marks

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

A1
Needed today
Solve $3 x^{2} + 6 x - 45 = 0$ .
[2 marks]
Answer
A2
Needed today
A rectangle has a perimeter of $40$ cm and a width of $x$ cm. Write an expression for its area in terms of $x$ .
[2 marks]
Answer
A3
From chapter 8
Work out the equation of the tangent to the curve $y = 4 + 3 x - x^{3}$ at the point where $x = 2$ .
[2 marks]
Answer
A4
Last lesson
$y = x^{3} - 5 x^{2} + 2 x$ . Work out the value of $\frac{d ^{2} y}{d x ^{2}}$ when $x = 3$ .
[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 coordinates of the stationary point on the curve $y = 2 x^{2} - 12 x + 7$ . Is it a maximum or a minimum?

Hint A stationary point is where

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

. Then look at the sign of

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

B1
Work out the coordinates of the stationary point on the curve $y = 5 + 8 x - 2 x^{2}$ . Is it a maximum or a minimum?
[4 marks]
Answer $(\dots\dots, \dots\dots)$ , a $\dots\dots\dots$

Example 2

Work out the coordinates of the stationary points on the curve $y = x^{3} - 6 x^{2} + 9 x + 1$ . Use $\frac{d ^{2} y}{d x ^{2}}$ to say whether each is a maximum or a minimum.

Not drawn accurately

Hint Set

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

and factorise. Put each

x

into

y

, then into

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

B2
Work out the coordinates of the stationary points on the curve $y = 2 x^{3} - 9 x^{2} - 24 x + 10$ . Use $\frac{d ^{2} y}{d x ^{2}}$ to say whether each is a maximum or a minimum.
[6 marks]

Example 3

$y = x^{4} - 8 x^{2} + 3$ . Work out the coordinates of the stationary points. Use the sign of the gradient on each side of each point to say whether it is a maximum or a minimum.

Hint Factorise

\frac{d y}{d x}

fully. Then choose a test value of

x

between each pair of stationary points.

B3
$y = 3 x^{4} - 4 x^{3} - 12 x^{2} + 1$ . Work out the coordinates of the three stationary points. Use the sign of the gradient on each side of each point to say whether it is a maximum or a minimum.
[6 marks]

Example 4

$y = 2 x + \frac{18}{x}$ for $x > 0$ . Work out the least value of $y$ . Show that it is a minimum.

Hint Write

y = 2 x + 18 x^{- 1}

. Use

x > 0

when you solve

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

B4
$y = x^{2} + \frac{128}{x}$ for $x > 0$ . Work out the least value of $y$ . Show that it is a minimum.
[5 marks]
Answer

Example 5

A farmer has $120$ m of fence to make a rectangular pen against a long wall. The wall is one side, so the fence makes the other three sides. The two sides at right angles to the wall are each $x$ m. Work out the greatest area the pen can have.

Hint Write the area

A

in terms of

x

only. Then solve

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

and check the sign of

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

B5
An open box is made from a square sheet of card with sides $24$ cm. A square of side $x$ cm is cut from each corner and the sides are folded up. Work out the value of $x$ that gives the greatest volume, and that volume. Show that it is a maximum.
[6 marks]
Answer $x = \dots\dots$ , volume $\dots\dots\dots$ cm³

Practice

Level 2

C1
The curve $y = a x^{2} + b x + c$ crosses the $y$ -axis at $(0, 11)$ and has a minimum point at $(2, 3)$ . Work out the values of $a$ , $b$ and $c$ .
[4 marks]
Answer $a = \dots\dots$ , $b = \dots\dots$ , $c = \dots\dots$
C2
The curve $y = p x + q x^{2}$ passes through the point $(2, 16)$ . Its gradient at that point is $4$ . Work out the values of $p$ and $q$ . Hence work out the maximum value of $y$ .
[5 marks]
Answer $p = \dots\dots$ , $q = \dots\dots$ , maximum $\dots\dots$

C3
$x$ and $y$ are positive numbers with $x + y = 18$ . $P = x^{2} y$ . Work out the greatest value of $P$ .
[4 marks]
Answer
C4
The curve $y = x^{3} + k x^{2} + 24 x - 5$ has a stationary point where $x = 2$ . Work out the value of $k$ . Hence work out the $x$ -coordinate of the other stationary point.
[4 marks]
Answer $k = \dots\dots$ , $x = \dots\dots$

Exam-style questions

AQA exam style

D1
Ella is finding the stationary points of $y = 4 x^{3} - 3 x$ . She writes: $\frac{d y}{d x} = 12 x^{2} - 3 = 0$ , so $x = \pm \frac{1}{2}$ . At $x = \frac{1}{2}$ , $\frac{d ^{2} y}{d x ^{2}} = 24 x = 12$ , which is positive, so $(\frac{1}{2}, - 1)$ is a maximum.
Explain Ella's mistake. Work out the coordinates of the maximum point.
[3 marks]
Answer $(\dots\dots, \dots\dots)$

D2
A curve has equation $y = x^{3} - 6 x^{2} - 15 x + 12$ .
(a)
Write down an expression for $\frac{d y}{d x}$ .
[1 mark]
Answer
(b)
Work out the coordinates of the stationary points and determine whether each is a maximum or a minimum. You must show your working.
[5 marks]
(c)
Sketch the curve on the axes. Label the stationary points and the point where the curve crosses the $y$ -axis.
[2 marks]
Total for D2: 8 marks
D3
The diagram shows an L shape. It is a rectangle $5 x$ cm by $y$ cm with a rectangle $3 x$ cm by $x$ cm cut from one corner. The perimeter of the L shape is $224$ cm.
Not drawn accurately
(a)
Show that $y = 112 - 5 x$ .
[2 marks]
(b)
The area of the L shape is $A$ cm². Show that $A = 560 x - 28 x^{2}$ .
[2 marks]
(c)
Use calculus to work out the maximum value of $A$ as $x$ varies.
[3 marks]
Answer $\dots\dots\dots$ cm²
Total for D3: 7 marks
D4
An open box has no lid. Its base is a square of side $x$ cm. Its height is $h$ cm. Its volume is $256$ cm³.
Not drawn accurately
(a)
Show that the area of card used to make the box, $A$ cm², is $A = x^{2} + \frac{1024}{x}$ .
[3 marks]
(b)
Work out the value of $x$ that makes $A$ as small as possible. Show that it gives a minimum.
[4 marks]
Answer $x = \dots\dots$
(c)
Work out the height of the box when $A$ is least.
[1 mark]
Answer $\dots\dots$ cm
Total for D4: 8 marks
D5
$y = 2 x + \frac{64}{x ^{2}}$ . Show that $y$ has a minimum value when $x = 4$ . Work out that minimum value.
[5 marks]
Answer $\dots\dots$

Extension

Stretch

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

E1
A metal tin is a cylinder with a base but no lid. Its volume is $64 π$ cm³. Work out the radius that uses the least metal, and the area of metal used. Give the area in terms of $π$ .
[6 marks]
Hint 1 · What to try
Write the volume and the area of metal in terms of $r$ and the height $h$ . Use the volume to remove $h$ .
Hint 2 · The first line
$h = \frac{64}{r ^{2}}$ , so the area is $π r^{2} + 2 π r \times \frac{64}{r ^{2}} = π r^{2} + \frac{128 π}{r}$ .
Hint 3 · The full method
$\frac{d A}{d r} = 2 π r - \frac{128 π}{r ^{2}} = 0$ gives $r^{3} = 64$ , so $r = 4$ . The second derivative is positive, so it is a minimum. The area is $16 π + 32 π = 48 π$ cm².
Answerradius $\dots\dots$ cm, area $\dots\dots$ cm²

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can work out the coordinates of the stationary points on a curve.
I can use the second derivative, or the gradient on each side, to say if a point is a maximum or a minimum.
I can sketch a curve from its stationary points and its y-intercept.
I can use calculus to find the greatest or least value in a problem.

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