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Date	May 2021	Marks available	3	Reference code	21M.1.SL.TZ1.8
Level	Standard Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 1
Command term	Sketch	Question number	8	Adapted from	N/A

Question

Let $y = \frac{\ln x}{x^{4}}$ for $x > 0$ .

Consider the function defined by $f (x) \frac{\ln x}{x^{4}}$ for $x > 0$ and its graph $y = f (x)$ .

Show that $\frac{d y}{d x} = \frac{1 - 4 \ln x}{x^{5}}$ .

[3]

The graph of $f$ has a horizontal tangent at point $P$ . Find the coordinates of $P$ .

[5]

Given that $f'' (x) = \frac{20 \ln x - 9}{x^{6}}$ , show that $P$ is a local maximum point.

[3]

Solve $f (x) > 0$ for $x > 0$ .

[2]

Sketch the graph of $f$ , showing clearly the value of the $x$ -intercept and the approximate position of point $P$ .

[3]

Markscheme

attempt to use quotient or product rule (M1)

$\frac{d y}{d x} = \frac{x^{4} (\frac{1}{x}) - (\ln x) (4 x^{3})}{{(x^{4})}^{2}}$ OR $(\ln x) (- 4 x^{- 5}) + (x^{- 4}) (\frac{1}{x})$ A1

correct working A1

$= \frac{x^{3} (1 - 4 \ln x)}{x^{8}}$ OR cancelling $x^{3}$ OR $\frac{- 4 \ln x}{x^{5}} + \frac{1}{x^{5}}$

$= \frac{1 - 4 \ln x}{x^{5}}$ AG

[3 marks]

$f' (x) = \frac{d y}{d x} = 0$ (M1)

$\frac{1 - 4 \ln x}{x^{5}} = 0$

$\ln x = \frac{1}{4}$ (A1)

$x = e^{\frac{1}{4}}$ A1

substitution of their $x$ to find $y$ (M1)

$y = \frac{\ln e^{\frac{1}{4}}}{{(e^{\frac{1}{4}})}^{4}}$

$= \frac{1}{4 e} (= \frac{1}{4} e^{- 1})$ A1

$P (e^{\frac{1}{4}}, \frac{1}{4 e})$

[5 marks]

$f'' (e^{\frac{1}{4}}) = \frac{20 \ln e^{\frac{1}{4}} - 9}{{(e^{\frac{1}{4}})}^{6}}$ (M1)

$= \frac{5 - 9}{e^{1.5}} (= - \frac{4}{e^{1.5}})$ A1

which is negative R1

hence $P$ is a local maximum AG

Note: The R1 is dependent on the previous A1 being awarded.

[3 marks]

$\ln x > 0$ (A1)

$x > 1$ A1

[2 marks]

A1A1A1

Note: Award A1 for one $x$ -intercept only, located at $1$

A1 for local maximum, $P$ , in approximately correct position
A1 for curve approaching $x$ -axis as $x \to \infty$ (including change in concavity).

[3 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » SL 2.3—Graphing

Show 53 related questions

22M.3.AHL.TZ1.2h.ii:
For $a = r$ and $b > 0$ , state in terms of $r$ , the coordinates of points $P$ and $A$ .
22M.3.AHL.TZ1.2h.i:
Sketch the curve $y = (x - r) (x^{2} - 2 a x + a^{2} + b^{2})$ for $a = r = 1$ and $b = 2$ .
22M.1.SL.TZ1.7d:
Sketch the graph of $f ″$ , the second derivative of $f$ . Indicate clearly the $x$ -intercept and the $y$ -intercept.
22M.1.SL.TZ2.4a.ii:
Write down the equation of the horizontal asymptote.
22M.1.SL.TZ2.4a.i:
Write down the equation of the vertical asymptote.
22M.1.SL.TZ2.4b:
On the set of axes below, sketch the graph of $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
22M.1.AHL.TZ2.3b:
On the set of axes below, sketch the graph of $y = f (x)$ .

On your sketch, clearly indicate the asymptotes and the position of any points of intersection with the axes.
18N.2.AHL.TZ0.H_9c.i:
sketch the graph of $y = f\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.2.AHL.TZ2.H_4a:
Sketch the graphs $y = {\text{si}}{{\text{n}}^3}\,x + {\text{ln}}\,x$ and $y = 1 + {\text{cos}}\,x$ on the following axes for 0 < $x$ ≤ 9.
18M.1.AHL.TZ2.H_2a:
Sketch the graphs of $y = \frac{x}{2} + 1$ and $y = \left| {x - 2} \right|$ on the following axes.
18N.2.SL.TZ0.T_4a:
Sketch the graph of y = f (x), for −4 ≤ x ≤ 3 and −50 ≤ y ≤ 100.
18N.2.SL.TZ0.T_4b.iii:
Use your graphic display calculator to find the equation of the tangent to the graph of y = f (x) at the point (–2, 38.75).

Give your answer in the form y = mx + c.
18N.2.SL.TZ0.T_4c:
Sketch the graph of the function g (x) = 10x + 40 on the same axes.
19M.2.SL.TZ2.T_5b:
Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .
21N.3.AHL.TZ0.1f:
Sketch the graph of $x^{2} - y^{2} = 1$ , stating the coordinates of any axis intercepts and the equation of each asymptote.
19N.1.SL.TZ0.S_10c:
The line ${L_2}$ is tangent to the graph of $g$ at ${\text{A}}$ and has equation $y = \left( {{\text{ln}}\,p} \right)x + q + 1$ .

The line ${L_2}$ passes through the point $\left( { - 2{\text{, }} - 2} \right)$ .

The gradient of the normal to $g$ at ${\text{A}}$ is $\frac{1}{{{\text{ln}}\left( {\frac{1}{3}} \right)}}$ .

Find the equation of ${L_1}$ in terms of $x$ .
19N.1.SL.TZ0.S_10a:
Write down the coordinates of ${\text{B}}$ .
18N.2.AHL.TZ0.H_9c.ii:
sketch the graph of $y = {f^{ - 1}}\left( x \right)$ , showing clearly any axis intercepts and giving the equations of any asymptotes.
19M.2.SL.TZ2.T_5c:
Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.
18M.2.AHL.TZ2.H_10a.i:
Sketch the graph of $y = f\left( x \right)$ for $- \frac{{5\pi }}{8} \leqslant x \leqslant \frac{\pi }{8}$ .
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.
22M.3.AHL.TZ2.1a.i:
$y^{2} = x^{3}, x \geq 0$
22M.3.AHL.TZ2.1a.ii:
$y^{2} = x^{3} + 1, x \geq - 1$
22M.1.SL.TZ2.4c:
Hence, solve the inequality $0 < \frac{2 x - 1}{x + 1} < 2$ .
22M.2.AHL.TZ2.10a.ii:
Plant $A$ correct to three significant figures.
18N.2.AHL.TZ0.H_9d:
Hence, or otherwise, solve the inequality $f\left( x \right) > {f^{ - 1}}\left( x \right)$ .
17N.2.SL.TZ0.T_5b.ii:
Find $f’(x)$ .
21M.3.AHL.TZ2.1b:
Use your graphic display calculator to explore the graph of $y = f_{n} (x)$ for

• the odd values $n = 3$ and $n = 5$ ;

• the even values $n = 2$ and $n = 4$ .

Hence, copy and complete the following table.
21M.3.AHL.TZ1.1a.ii:
$c = 2$ .
21N.2.SL.TZ0.2b:
Sketch the graph of $f$ on the following grid.
17N.1.AHL.TZ0.H_6a:
Sketch the graph of $y = \frac{{1 - 3x}}{{x - 2}}$ , showing clearly any asymptotes and stating the coordinates of any points of intersection with the axes.
19N.1.SL.TZ0.S_10b:
Given that $f'\left( a \right) = \frac{1}{{{\text{ln}}\,p}}$ , find the equation of ${L_1}$ in terms of $x$ , $p$ and $q$ .
19N.1.SL.TZ0.S_3b:
The graph of $f\left( x \right) = {x^2}$ is transformed to obtain the graph of $g$ .

Describe this transformation.
21N.3.AHL.TZ0.1b:
Show that ${(f (t))}^{2} + {(g (t))}^{2} = f (2 t)$ .
21N.3.AHL.TZ0.1c.i:
$f (i u)$ .
21N.3.AHL.TZ0.1g:
The hyperbola with equation $x^{2} - y^{2} = 1$ can be rotated to coincide with the curve defined by $x y = k, k \in ℝ$ .

Find the possible values of $k$ .
22M.2.AHL.TZ1.10a:
Sketch the curve $y = f (x)$ , clearly indicating the coordinates of the endpoints.
22M.1.AHL.TZ2.11a:
Sketch the curve $y = f (x)$ , clearly indicating any asymptotes with their equations. State the coordinates of any local maximum or minimum points and any points of intersection with the coordinate axes.
22M.2.AHL.TZ2.10a.i:
Plant $B$ .
21N.3.AHL.TZ0.1a:
Verify that $u = f (t)$ satisfies the differential equation $\frac{d^{2} u}{d t^{2}} = u$ .
21N.3.AHL.TZ0.1e:
Show that ${(f (t))}^{2} - {(g (t))}^{2} = {(f (i u))}^{2} - {(g (i u))}^{2}$ .
17N.2.SL.TZ0.T_5b.i:
Expand the expression for $f(x)$ .
21N.2.SL.TZ0.2a:
Find the values of $x$ for which $f (x) = 0$ .
19N.1.SL.TZ0.S_3a:
Find the value of $b$ .
21M.3.AHL.TZ2.1a:
Sketch the graph of $y = f_{1} (x)$ , stating the values of any axes intercepts and the coordinates of any local maximum or minimum points.
21M.3.AHL.TZ1.1a.i:
$c = 1$ .
18N.2.AHL.TZ0.H_9b:
Hence, or otherwise, find the coordinates of the point of inflexion on the graph of $y = f\left( x \right)$ .
19M.2.SL.TZ2.T_5h:
Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .
17N.2.SL.TZ0.T_5e:
Write down the coordinates of the point of intersection.
21N.3.AHL.TZ0.1c.ii:
$g (i u)$ .
21N.3.AHL.TZ0.1d:
Hence find, and simplify, an expression for ${(f (i u))}^{2} + {(g (i u))}^{2}$ .
18M.2.SL.TZ1.T_4e:
Sketch the graph of y = f (x) for 0 < x ≤ 6 and −30 ≤ y ≤ 60.
Clearly indicate the minimum point P and the x-intercepts on your graph.
17N.2.SL.TZ0.T_5d:
Draw the graph of $f$ for $- 3 \leqslant x \leqslant 3$ and $- 40 \leqslant y \leqslant 20$ . Use a scale of 2 cm to represent 1 unit on the $x$ -axis and 1 cm to represent 5 units on the $y$ -axis.

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