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Date	May 2013	Marks available	2	Reference code	13M.1.sl.TZ1.15
Level	SL only	Paper	1	Time zone	TZ1
Command term	Find	Question number	15	Adapted from	N/A

Question

Consider the function \(f (x) = ax^3 − 3x + 5\), where \(a \ne 0\).

Find \(f ' (x) \).

[2]

Write down the value of \(f ′(0)\).

[1]

The function has a local maximum at x = −2.

Calculate the value of a.

[3]

Markscheme

\( f '(x) = 3ax^2 - 3\) (A1)(A1) (C2)

Note: Award a maximum of (A1)(A0) if any extra terms are seen.

−3 (A1)(ft) (C1)

Note: Follow through from their part (a).

\(f '(x) = 0\) (M1)

Note: This may be implied from line below.

\(3a(-2)^2 - 3 = 0\) (M1)

\((a =) \frac{1}{4}\) (A1)(ft) (C3)

Note: Follow through from their part (a).

Examiners report

Many candidates could find the derivative of the cubic function and find the value of the derivative at \(x = 0\).

Many candidates could find the derivative of the cubic function and find the value of the derivative at \(x = 0\). For part (c) many candidates calculated the value of the function rather than the derivative at \(x = - 2\). However only the best realized that the derivative is zero at the maximum and so calculated the value of \(a\).

Syllabus sections

Topic 7 - Introduction to differential calculus » 7.2 » The derivative of functions of the form \(f\left( x \right) = a{x^n} + b{x^{n - 1}} + \ldots \), where all exponents are integers.

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