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Date	May Example questions	Marks available	4	Reference code	EXM.1.AHL.TZ0.6
Level	Additional Higher Level	Paper	Paper 1 (without calculator)	Time zone	Time zone 0
Command term	Find	Question number	6	Adapted from	N/A

Question

Let $f\left( x \right) = \frac{{{x^2} - 10x + 5}}{{x + 1}}{\text{,}}\,\,x \in \mathbb{R}{\text{,}}\,\,x \ne - 1$ .

Find the co-ordinates of all stationary points.

[4]

Write down the equation of the vertical asymptote.

[1]

With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.

[4]

Markscheme

$f'\left( x \right) = \frac{{\left( {2x - 10} \right)\left( {x + 1} \right) - \left( {{x^2} - 10x + 5} \right)1}}{{{{\left( {x + 1} \right)}^2}}}$ M1

$f'\left( x \right) = 0 \Rightarrow {x^2} + 2x - 15 = 0 \Rightarrow \left( {x + 5} \right)\left( {x - 3} \right) = 0$ M1

Stationary points are $\left( { - 5{\text{,}}\,\, - 20} \right)\,\,{\text{and}}\,\,\left( {3{\text{,}}\,\, - 4} \right)$ A1A1

[4 marks]

$x = - 1$ A1

[1 mark]

Looking at the nature table

M1A1

$\left( { - 5{\text{,}}\, - 20} \right)$ is a max and $\left( {3{\text{,}}\, - 4} \right)$ is a min A1A1

[4 marks]

Examiners report

[N/A]

Syllabus sections

Topic 2—Functions » AHL 2.13—Rational functions

Show 40 related questions

21N.2.AHL.TZ0.10d:
Sketch the graph of $f$ for $- 30 \leq x \leq 30$ , clearly indicating the points of intersection with each axis and any asymptotes.
EXM.1.AHL.TZ0.5b:
Write down the equation of the vertical asymptote.
EXM.1.AHL.TZ0.6c:
With justification, state if each stationary point is a minimum, maximum or horizontal point of inflection.
EXM.1.AHL.TZ0.4b:
Find the equation of the horizontal asymptote.
EXM.1.AHL.TZ0.4c:
Find the exact value of $\int\limits_0^1 {f\left( x \right)} \,dx$ , giving the answer in the form ${\text{ln}}\,q{\text{,}}\,\,q \in \mathbb{Q}$ .
18N.2.AHL.TZ0.H_11a:
Find an expression for ${\theta _p}$ in terms of $p$ .
18N.2.AHL.TZ0.H_11b:
Show that $p = \frac{{{q^2} + q - 1}}{{2q + 1}}$ .
19N.1.AHL.TZ0.H_10b.i:
find the coordinates of the $y$ -intercept.
19N.1.AHL.TZ0.H_10b.ii:
show that there are no $x$ -intercepts.
19N.1.AHL.TZ0.H_10b.iii:
sketch the graph, showing clearly any asymptotic behaviour.
EXM.1.AHL.TZ0.6b:
Write down the equation of the vertical asymptote.
18N.2.AHL.TZ0.H_11c:
By sketching the graph of $p$ as a function of $q$ , determine the range of values of $p$ for which there are possible values of $q$ .
17N.2.AHL.TZ0.H_6b:
Find the expected number of weeks in the year in which Lucca eats no bananas.
22M.2.AHL.TZ1.6b:
Solve the inequality $f (x) \geq g (x)$ .
20N.1.AHL.TZ0.H_10a.i:
Write down an expression for $f' (x)$ .
20N.1.AHL.TZ0.H_10b.i:
Show that $g^{- 1}$ exists.
SPM.2.AHL.TZ0.9:
Consider the graphs of $y = \frac{{{x^2}}}{{x - 3}}$ and $y = m\left( {x + 3} \right)$ , $m \in \mathbb{R}$ .

Find the set of values for $m$ such that the two graphs have no intersection points.
21M.2.AHL.TZ1.11d:
Sketch the graph of $y = f (x)$ for $- 3 \leq x \leq 3$ , showing the values of any axes intercepts, the coordinates of any local maxima and local minima, and giving the equations of any asymptotes.
21M.2.AHL.TZ1.11c:
Find the $x$ -coordinate of the point of inflexion.
21M.2.AHL.TZ1.11a:
Find the value of $p$ and the value of $q$ .
20N.1.AHL.TZ0.H_10b.iii:
Hence find $g^{- 1} (x)$ .
20N.1.AHL.TZ0.H_10a.ii:
Hence, given that $f^{- 1}$ does not exist, show that $b^{2} - 3 a c > 0$ .
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.
EXM.1.AHL.TZ0.5c:
As $x \to \pm \infty$ the graph of $f\left( x \right)$ approaches an oblique straight line asymptote.

Divide $2{x^2} - 5x - 12$ by $x + 2$ to find the equation of this asymptote.
17N.2.AHL.TZ0.H_6a:
Find the probability that Lucca eats at least one banana in a particular day.
21N.2.AHL.TZ0.10a.ii:
$y$ -axis.
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.
21N.2.AHL.TZ0.10c:
The oblique asymptote of the graph of $f$ can be written as $y = a x + b$ where $a, b \in ℚ$ .

Find the value of $a$ and the value of $b$ .
EXM.1.AHL.TZ0.4a:
Show that $f\left( x \right)$ has no vertical asymptotes.
20N.1.AHL.TZ0.H_10b.ii:
$g (x)$ can be written in the form $p (x - 2)^{3} + q$ , where $p, q \in ℝ$ .

Find the value of $p$ and the value of $q$ .
20N.1.AHL.TZ0.H_10d:
Sketch the graphs of $y = g (x)$ and $y = g^{- 1} (x)$ on the same set of axes, indicating the points where each graph crosses the coordinate axes.
21M.2.AHL.TZ1.11b:
Find an expression for $f' (x)$ .
EXM.1.AHL.TZ0.5a:
Find all the intercepts of the graph of $f\left( x \right)$ with both the $x$ and $y$ axes.
21M.2.AHL.TZ1.11e:
Find the equations of all the asymptotes on the graph of $y = g (x)$ .
20N.1.AHL.TZ0.H_10c:
State each of the transformations in the order in which they are applied.
21M.2.AHL.TZ1.11f:
By considering the graph of $y = g (x) - f (x)$ , or otherwise, solve $f (x) < g (x)$ for $x \in ℝ$ .
21N.2.AHL.TZ0.10e.i:
Express $\frac{1}{f (x)}$ in partial fractions.
21N.2.AHL.TZ0.10e.ii:
Hence find the exact value of $\int_{0}^{3} \frac{1}{f (x)} d x$ , expressing your answer as a single logarithm.
21N.2.AHL.TZ0.10a.i:
$x$ -axis.
21N.2.AHL.TZ0.10b:
Write down the equation of the vertical asymptote of the graph of $f$ .

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