A-level · Differentiation

Stationary points (maxima and minima)Exam-style questions

Finding where a curve turns, and using it to find the greatest or least value of something. What a stationary point is and why dy/dx = 0 finds it; full coordinates and which roots to keep; telling a maximum from a minimum with the second derivative; what to do when the second derivative is zero; where a curve is increasing or decreasing; and optimisation questions in context.

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25 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4 Level 5

1
Figure 5 shows an open-topped storage tank in the shape of a cuboid. The base is a square of side $x$ metres and the height of the tank is $h$ metres.Figure 5 (not accurately drawn)

The tank must hold exactly $32$ m³ of liquid. The base is made from a reinforced material costing £ $15$ per m², and the four vertical sides are made from a lighter material costing £ $6$ per m².
(a) Show that the total cost, £ $C$ , of the material used to make the tank is given by

$C = 15x^2 + \dfrac{768}{x}$

(4 marks)
(b) Use calculus to find the value of $x$ for which $C$ is a minimum, giving your answer to $3$ significant figures, and prove that your value gives a minimum. (5 marks)
(c) Hence find the minimum cost of the material, to the nearest pound. (1 mark)
Level 4 · 10 marksTry it
2
The curve $C$ has equation $y = x^3 e^{-x}$ .
(a) Show that $\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^2 e^{-x}(3 - x)$ . (2 marks)
(b) Find the coordinates of the stationary points of $C$ . (2 marks)
(c) Determine the nature of each stationary point. (3 marks)
Level 4 · 7 marksTry it
3
A rectangle $ABCD$ is drawn under the curve with equation $y = 27 - 3x^2$ , as shown in the diagram below.The side $AB$ lies on the $x$ -axis and the vertices $C$ and $D$ lie on the curve.

$B$ is the point $(q, 0)$
(a) Show that the area, $A$ , of rectangle $ABCD$ is given by

$A = 54q - 6q^3 \quad \text{for } 0 < q < k$

where $k$ is a constant to be found.

(3 marks)
(b) Find the exact maximum area of rectangle $ABCD$

Fully justify your answer.

(6 marks)
Level 4 · 9 marksTry it
4
The diagram below shows the curve with equation $y = 12 - 2x^2$ and the triangle $ORS$ , where $O$ is the origin.The vertices $R$ and $S$ lie on the curve and the side $RS$ is horizontal.

$S$ has $x$ -coordinate $q$
(a) Show that the area, $A$ , of triangle $ORS$ is given by

$A = 12q - 2q^3 \quad \text{for } 0 < q < k$

where $k$ is a constant to be found.

(3 marks)
(b) Find the exact maximum area of triangle $ORS$

Fully justify your answer.

(6 marks)
Level 4 · 9 marksTry it
5
The curve $C$ has equation

$y = \dfrac{x^3 - 8x^2 + 28x + 3}{\sqrt{x}} \qquad x > 0$
(a) Write $y$ in the form $x^{\frac{5}{2}} + px^{\frac{3}{2}} + qx^{\frac{1}{2}} + rx^{-\frac{1}{2}}$ , where $p$ , $q$ and $r$ are constants to be found.

(2 marks)
(b) Use calculus to show that the $x$ coordinate of each turning point of $C$ is a solution of the equation

$5x^3 - 24x^2 + 28x - 3 = 0$

(4 marks)

Given that $(x - 3)$ is a factor of $5x^3 - 24x^2 + 28x - 3$
(c) use algebra to find, in exact form, the $x$ coordinates of the remaining two turning points of $C$ .

(3 marks)
Level 4 · 9 marksTry it

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