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Date	May Specimen	Marks available	2	Reference code	SPM.1.sl.TZ0.14
Level	SL only	Paper	1	Time zone	TZ0
Command term	Write down	Question number	14	Adapted from	N/A

Question

A sketch of the function \(f(x) = 5{x^3} - 3{x^5} + 1\) is shown for \( - 1.5 \leqslant x \leqslant 1.5\) and \( - 6 \leqslant y \leqslant 6\) .

Write down \(f'(x)\) .

[2]

Find the equation of the tangent to the graph of \(y = f(x)\) at \((1{\text{, }}3)\) .

[2]

Write down the coordinates of the second point where this tangent intersects the graph of \(y = f(x)\) .

[2]

Markscheme

\(f'(x) = 15{x^2} - 15{x^4}\) (A1)(A1) (C2)

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

\(f'(1) = 0\) (M1)

Note: Award (M1) for \(f'(x) = 0\) .

\(y = 3\) (A1)(ft) (C2)

Note: Follow through from their answer to part (a).

\(( - 1.38{\text{, }}3)\) \(( - 1.38481 \ldots {\text{, }}3)\) (A1)(ft)(A1)(ft) (C2)

Note: Follow through from their answer to parts (a) and (b).

Note: Accept \(x = - 1.38\), \(y = 3\) (\(x = - 1.38481 \ldots\) , \(y = 3\)) .

Examiners report

[N/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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