17M.1.AHL.TZ1.H_12

pestleMathematicsAAHLPaper 117M· sl-5-7-the-second-derivativesource ↗

Consider the function $q\left( x \right) = {x^5} - 10{x^2} + 15x - 6,{\text{ }}x \in \mathbb{R}$ .

Show that the graph of $y = q(x)$ is concave up for $x > 1$ .

[3]

e.i.

Sketch the graph of $y = q(x)$ showing clearly any intercepts with the axes.

[3]

e.ii.

Markscheme / solution

$\frac{{{{\text{d}}^2}y}}{{{\text{d}}{x^2}}} = 20{x^3} - 20$ M1A1

for $x > 1,{\text{ }}20{x^3} - 20 > 0 \Rightarrow$ concave up R1AG

[3 marks]

e.i.

M17/5/MATHL/HP1/ENG/TZ1/B12.e.ii/M

$x$ -intercept at $(1,{\text{ }}0)$ A1

$y$ -intercept at $(0,{\text{ }} - 6)$ A1

stationary point of inflexion at $(1,{\text{ }}0)$ with correct curvature either side A1

[3 marks]

e.ii.

Examiners’ report

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e.i.

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e.ii.