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Date	May 2018	Marks available	4	Reference code	18M.1.AHL.TZ1.H_9
Level	Additional Higher Level	Paper	Paper 1	Time zone	Time zone 1
Command term	Sketch	Question number	H_9	Adapted from	N/A

Question

Let $f\left( x \right) = \frac{{2 - 3{x^5}}}{{2{x^3}}},\,\,x \in \mathbb{R},\,\,x \ne 0$ .

The graph of $y = f\left( x \right)$ has a local maximum at A. Find the coordinates of A.

[5]

a.

Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .

[5]

b.i.

The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.

[3]

b.ii.

Sketch the graph of $y = f\left( x \right)$ showing clearly the position of the points A and B.

[4]

c.

Markscheme

* This question is from an exam for a previous syllabus, and may contain minor differences in marking or structure.

attempt to differentiate (M1)

$f'\left( x \right) = - 3{x^{ - 4}} - 3x$ A1

Note: Award M1 for using quotient or product rule award A1 if correct derivative seen even in unsimplified form, for example $f'\left( x \right) = \frac{{ - 15{x^4} \times 2{x^3} - 6{x^2}\left( {2 - 3{x^5}} \right)}}{{{{\left( {2{x^3}} \right)}^2}}}$ .

$- \frac{3}{{{x^4}}} - 3x = 0$ M1

$\Rightarrow {x^5} = - 1 \Rightarrow x = - 1$ A1

${\text{A}}\left( { - 1,\, - \frac{5}{2}} \right)$ A1

[5 marks]

a.

$f''\left( x \right) = 0$ M1

$f''\left( x \right) = 12{x^{ - 5}} - 3\left( { = 0} \right)$ A1

Note: Award A1 for correct derivative seen even if not simplified.

$\Rightarrow x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)$ A1

hence (at most) one point of inflexion R1

Note: This mark is independent of the two A1 marks above. If they have shown or stated their equation has only one solution this mark can be awarded.

$f''\left( x \right)$ changes sign at $x = \sqrt[5]{4}\left( { = {2^{\frac{2}{5}}}} \right)$ R1

so exactly one point of inflexion

[5 marks]

b.i.

$x = \sqrt[5]{4} = {2^{\frac{2}{5}}}\left( { \Rightarrow a = \frac{2}{5}} \right)$ A1

$f\left( {{2^{\frac{2}{5}}}} \right) = \frac{{2 - 3 \times {2^2}}}{{2 \times {2^{\frac{6}{5}}}}} = - 5 \times {2^{ - \frac{6}{5}}}\left( { \Rightarrow b = - 5} \right)$ (M1)A1

Note: Award M1 for the substitution of their value for $x$ into $f\left( x \right)$ .

[3 marks]

b.ii.

A1A1A1A1

A1 for shape for x < 0
A1 for shape for x > 0
A1 for maximum at A
A1 for POI at B.

Note: Only award last two A1s if A and B are placed in the correct quadrants, allowing for follow through.

[4 marks]

c.

Examiners report

[N/A]

a.

[N/A]

b.i.

[N/A]

b.ii.

[N/A]

c.

Syllabus sections

Topic 2—Functions » SL 2.2—Functions, notation domain, range and inverse as reflection

Show 123 related questions

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17M.1.SL.TZ2.T_14a:
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19M.2.SL.TZ1.T_4b.ii:
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19M.1.AHL.TZ1.H_8d:
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18M.2.AHL.TZ2.H_10a.ii:
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18M.2.AHL.TZ2.H_10a.i:
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18M.2.SL.TZ2.T_6b:
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19M.2.SL.TZ1.S_9d.i:
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18N.1.AHL.TZ0.H_3c.ii:
For the value of $a$ found in part (b), find an expression for ${g^{ - 1}}\left( x \right)$ .
18M.2.SL.TZ1.T_4b:
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18M.1.AHL.TZ2.H_10b.i:
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18M.2.AHL.TZ2.H_10a.iii:
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19M.2.SL.TZ1.S_9c:
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17M.1.AHL.TZ1.H_11e:
Sketch the graph of $y = f\left( {\left| x \right|} \right)$ .
17M.2.AHL.TZ1.H_12c:
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16N.1.AHL.TZ0.H_3b:
find the value of $b$ .
18M.1.SL.TZ1.S_3c:
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16N.2.AHL.TZ0.H_5b:
State the range of $f$ .
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17M.2.SL.TZ2.S_6b:
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17M.1.AHL.TZ2.H_2b:
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17M.1.AHL.TZ1.H_11d:
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17N.2.AHL.TZ0.H_10a.i:
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19M.1.AHL.TZ1.H_8a:
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17M.1.SL.TZ2.T_14c:
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19M.2.SL.TZ1.T_4e:
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18M.1.AHL.TZ2.H_10c:
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State the domain and range of $h \circ g$ .
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18M.1.AHL.TZ1.H_9b.i:
Show that there is exactly one point of inflexion, B, on the graph of $y = f\left( x \right)$ .
18M.1.AHL.TZ2.H_2b:
Solve the equation $\frac{x}{2} + 1 = \left| {x - 2} \right|$ .
18M.1.AHL.TZ1.H_9a:
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17M.1.AHL.TZ1.H_11a.i:
Express ${x^2} + 3x + 2$ in the form ${(x + h)^2} + k$ .
16N.1.AHL.TZ0.H_3a:
state the value of $a$ and the value of $c$ ;
17N.2.AHL.TZ0.H_10c:
Find the coordinates of the point on the graph of $f$ where the normal to the graph is parallel to the line $y = - x$ .
17N.2.AHL.TZ0.H_10b:
Sketch the graph of $y = f(x)$ showing clearly the minimum point and any asymptotic behaviour.
20N.1.SL.TZ0.T_2a.i:
State, in the context of the question, what the value of $34.50$ represents.
20N.1.SL.TZ0.T_2a.ii:
State, in the context of the question, what the value of $8.50$ represents.
20N.1.SL.TZ0.T_2c:
Kaelani has $450 PGK$ .

Find the maximum number of large cheese pizzas that Kaelani can order from Olava’s Pizza Company.
17N.2.AHL.TZ0.H_10a.ii:
Determine the values of $x$ for which $f(x)$ is a decreasing function.
17M.1.AHL.TZ1.H_11c:
Show that $\frac{1}{{x + 1}} - \frac{1}{{x + 2}} = \frac{1}{{{x^2} + 3x + 2}}$ .
20N.1.SL.TZ0.T_2b:
Write down the minimum number of pizzas that can be ordered.
17M.1.SL.TZ1.T_12a:
Write down the domain of the function.
16N.2.AHL.TZ0.H_5c:
Solve the inequality $\left| {3x\arccos (x)} \right| > 1$ .
17M.2.AHL.TZ1.H_12f:
Find $g'(x)$ .
16N.2.SL.TZ0.S_1a:
Find $f(8)$ .
17M.2.SL.TZ2.S_6c:
The equation $(f \circ g)(x) = k$ has exactly two solutions, for $0 \leqslant x \leqslant 2.25$ . Find the possible values of $k$ .
EXN.2.SL.TZ0.3c:
Find the range of $h$ .
18M.2.SL.TZ2.T_6d:
Find $\frac{{{\text{dy}}}}{{{\text{dx}}}}$ .
18M.1.AHL.TZ1.H_9b.ii:
The coordinates of B can be expressed in the form B $\left( {{2^a},\,b \times {2^{ - 3a}}} \right)$ where a, b $\in \mathbb{Q}$ . Find the value of a and the value of b.
17M.2.AHL.TZ1.H_12d:
Explain why the inverse function ${f^{ - 1}}$ does not exist.
17M.2.AHL.TZ1.H_12g.i:
Hence, show that there are no solutions to $g'(x) = 0$ ;
17N.1.AHL.TZ0.H_6b:
Hence or otherwise, solve the inequality $\left| {\frac{{1 - 3x}}{{x - 2}}} \right| < 2$ .
21M.1.SL.TZ1.4a:
Find the total cost of buying $40$ litres of gas at Leon’s gas station.
18M.2.SL.TZ1.T_4c:
Use your answer to part (b) to show that the minimum value of f(x) is −22 .
21M.1.SL.TZ1.4b:
Find $L^{- 1} (70)$ .
18M.2.AHL.TZ2.H_10d.i:
Find $\alpha$ and β in terms of $k$ .
19M.2.SL.TZ2.T_5d:
Find $f'\left( x \right)$ .
17M.1.SL.TZ1.T_12b.i:
Draw the line $y = - 6$ on the axes.
19M.2.SL.TZ1.T_4d:
Find the distance from the centre of Orangeton to the point at which the road meets the highway.
19M.1.AHL.TZ1.H_8b:
find the value of $f\left( 1 \right)$ .
19N.1.SL.TZ0.T_10b:
While in Kota Kinabalu, Criselda spent $440 MYR$ . She returned to the Currency Exchange counter and changed the remainder of her $MYR$ into $USD$ .

Calculate the amount of $USD$ she received.
17M.2.SL.TZ2.S_6a:
Show that $(f \circ g)(x) = {x^4} - 4{x^2} + 3$ .
19M.2.AHL.TZ2.H_4b:
Hence solve ${\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x - {\text{cos}}\,x - 1 < 0$ in the range 0 < $x$ ≤ 9.
17N.1.SL.TZ0.S_5b:
Given that $\mathop {\lim }\limits_{x \to + \infty } (g \circ f)(x) = - 3$ , find the value of $b$ .
16N.2.SL.TZ0.S_1c:
Solve $(g \circ f)(x) = 0$ .
18M.1.AHL.TZ2.H_10a:
Find the inverse function ${f^{ - 1}}$ , stating its domain.
19M.2.SL.TZ1.S_9a:
Find the gradient of $L$ .
19N.1.SL.TZ0.T_10a:
Calculate the amount of $MYR$ that Criselda received.
19M.2.SL.TZ1.S_9d.ii:
Hence, write down ${f^{ - 1}}\left( x \right)$ .
18M.2.AHL.TZ2.H_10a.iv:
Explain why $f$ is not a function for $- \frac{{3\pi }}{4} \leqslant x \leqslant \frac{\pi }{4}$ .
16N.2.AHL.TZ0.H_5a:
Sketch the graph of $f$ indicating clearly any intercepts with the axes and the coordinates of any local maximum or minimum points.
17M.1.SL.TZ1.T_12c:
Find the range of values of $k$ for which $f(x) = k$ has no solution.
17M.2.AHL.TZ1.H_12b:
Sketch the graph of $y = f(x)$ showing clearly the equations of asymptotes and the coordinates of any intercepts with the axes.
17M.1.SL.TZ1.T_12b.ii:
Write down the number of solutions to $f(x) = - 6$ .
21M.1.SL.TZ2.7b:
Find the value of $f^{- 1} (0)$ .
21M.1.SL.TZ2.7a:
Find the range of $f$ .
21M.1.AHL.TZ2.2c:
Write down the range of $f^{- 1} (x)$ .
17M.2.SL.TZ2.S_3a:
Write down the range of $f$ .
17M.2.AHL.TZ1.H_12a:
Find the largest possible domain $D$ for $f$ to be a function.
19M.1.AHL.TZ1.H_8c:
find the value of $f\left( 4 \right)$ .
21M.1.AHL.TZ2.2b:
Find an expression for the inverse function $f^{- 1} (x)$ . The domain is not required.
21M.1.AHL.TZ2.2a:
Find the range of $f$ .
18M.2.AHL.TZ2.H_10b:
Show that $g\left( t \right) = {\left( {\frac{{1 + t}}{{1 - t}}} \right)^2}$ .
17M.1.AHL.TZ1.H_11f:
Determine the area of the region enclosed between the graph of $y = f\left( {\left| x \right|} \right)$ , the $x$ -axis and the lines with equations $x = - 1$ and $x = 1$ .
17M.1.AHL.TZ1.H_11b:
Sketch the graph of $f(x)$ , indicating on it the equations of the asymptotes, the coordinates of the $y$ -intercept and the local maximum.
19M.2.SL.TZ2.T_5e:
Find the gradient of the graph of $y = f\left( x \right)$ at $x = 2$ .
EXN.2.SL.TZ0.3b.i:
Find the value of $t$ at which the ball hits the ground.
EXN.2.SL.TZ0.3a:
Find an expression for the height $h$ of the ball at time $t$ .
21N.1.SL.TZ0.5b.i:
Find $h^{- 1} (10)$ .
21N.1.SL.TZ0.5b.ii:
In the context of the question, interpret your answer to part (b)(i).
21N.1.SL.TZ0.5b.iii:
Write down the range of $h^{- 1}$ .

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