22M.1.AHL.TZ1.17

pestleMathematicsAIHLPaper 122M· sl-5-6-stationary-points-local-max-and-minsource ↗

A function $f$ is of the form $f (t) = p e^{q \cos (r t)}, p, q, r \in ℝ^{+}$ . Part of the graph of $f$ is shown.

The points $A$ and $B$ have coordinates $A (0, 6.5)$ and $B (5.2, 0.2)$ , and lie on $f$ .

The point $A$ is a local maximum and the point $B$ is a local minimum.

Find the value of $p$ , of $q$ and of $r$ .

Markscheme / solution

substitute coordinates of $A$

$f (0) = p e^{q \cos (0)} = 6.5$

$6.5 = p e^{q}$ (A1)

substitute coordinates of $B$

$f (5.2) = p e^{q \cos (5.2 r)} = 0.2$

EITHER

$f' (t) = - p q r \sin (r t) e^{q \cos (r t)}$ (M1)

minimum occurs when $- p q r \sin (5.2 r) e^{q \cos (5.2 r)} = 0$

$\sin (r t) = 0$

$r \times 5.2 = π$ (A1)

minimum value occurs when $\cos (r t) = - 1$ (M1)

$r \times 5.2 = π$ (A1)

period $= 2 \times 5.2 = 10.4$ (A1)

$r = \frac{2 π}{10.4}$ (M1)

THEN

$r = \frac{π}{5.2} = 0.604152 \dots (0.604)$ A1

$0.2 = p e^{- q}$ (A1)

eliminate $p$ or $q$ (M1)

$e^{2 q} = \frac{6.5}{0.2}$ OR $0.2 = \frac{p^{2}}{6.5}$

$q = 1.74 (1.74062 \dots)$ A1

$p = 1.14017 \dots (1.14)$ A1

[8 marks]

Examiners’ report

This was a challenging question and suitably positioned at the end of the examination. Candidates who attempted it were normally able to substitute points A and B into the given equation. Some were able to determine the first derivative. Only a few candidates were able to earn significant marks for this question.