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Date	May 2019	Marks available	1	Reference code	19M.2.SL.TZ2.T_5
Level	Standard Level	Paper	Paper 2	Time zone	Time zone 2
Command term	Write down	Question number	T_5	Adapted from	N/A

Question

Consider the function $f\left( x \right) = \frac{1}{3}{x^3} + \frac{3}{4}{x^2} - x - 1$ .

The function has one local maximum at $x = p$ and one local minimum at $x = q$ .

Write down the $y$ -intercept of the graph of $y = f\left( x \right)$ .

[1]

b.

Sketch the graph of $y = f\left( x \right)$ for −3 ≤ $x$ ≤ 3 and −4 ≤ $y$ ≤ 12.

[4]

c.

Determine the range of $f\left( x \right)$ for $p$ ≤ $x$ ≤ $q$ .

[3]

h.

Markscheme

−1 (A1)

Note: Accept (0, −1).

[1 mark]

b.

(A1)(A1)(A1)(A1)

Note: Award (A1) for correct window and axes labels, −3 to 3 should be indicated on the $x$ -axis and −4 to 12 on the $y$ -axis.
(A1)) for smooth curve with correct cubic shape;
(A1) for $x$ -intercepts: one close to −3, the second between −1 and 0, and third between 1 and 2; and $y$ -intercept at approximately −1;
(A1) for local minimum in the 4th quadrant and maximum in the 2nd quadrant, in approximately correct positions.
Graph paper does not need to be used. If window not given award at most (A0)(A1)(A0)(A1).

[4 marks]

c.

$- 1.27 \leqslant {\text{ }}f\left( x \right) \leqslant 1.33\,\,\,\left( { - 1.27083 \ldots \leqslant {\text{ }}f\left( x \right) \leqslant 1.33333 \ldots {\text{,}}\,\, - \frac{{61}}{{48}} \leqslant {\text{ }}f\left( x \right) \leqslant \frac{4}{3}} \right)$ (A1)(ft)(A1)(ft)(A1)

Note: Award (A1) for −1.27 seen, (A1) for 1.33 seen, and (A1) for correct weak inequalities with their endpoints in the correct order. For example, award (A0)(A0)(A0) for answers like $5 \leqslant {\text{ }}f\left( x \right) \leqslant 2$ . Accept $y$ in place of $f\left( x \right)$ . Accept alternative correct notation such as [−1.27, 1.33].

Follow through from their $p$ and $q$ values from part (g) only if their $f\left( p \right)$ and $f\left( q \right)$ values are between −4 and 12. Award (A0)(A0)(A0) if their values from (g) are given as the endpoints.

[3 marks]

h.

Examiners report

[N/A]

b.

[N/A]

c.

[N/A]

h.

Syllabus sections

Topic 5 —Calculus » SL 5.1—Introduction of differential calculus

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