EXM.1.AHL.TZ0.6

pestleMathematicsAAHLPaper 1EXM· ahl-2-13-rational-functionssource ↗

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 / solution

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