First slide

Maxima and minima

Question

The difference between the greatest and least values of the function $f (x) = \cos x + \frac{1}{2} \cos 2 x - \frac{1}{3} \cos 3 x i s$

Moderate

A

$2 / 3$
B

$8 / 7$
C

$9 / 4$
D

$3 / 8$

Solution

The given function is periodic, with period $2 π$ . So the difference between the greatest and least values of the
function is the difference between these values on the interval $[0, 2 π]$ .
We have
$\begin{aligned} f^{'} (x) = - (\sin x + \sin 2 x - \sin 3 x) \\ = - 4 \sin x \sin (3 x / 2) \sin (x / 2) \end{aligned}$
$Hence x = 0, 2 π / 3, π and 2 π are the critical points. Also,$
$f (0) = 1 + 1 / 2 - 1 / 3 = 7 / 6, f (2 π / 3) = - 13 / 12, f (π) = - 1 / 6 a n d f (2 π) = 7 / 6.$
Hence the greatest value is 7/6 and the least value is $- 13 / 12$ .
Thus the difference is $\frac{7}{6} - (- \frac{13}{12}) = \frac{27}{12} = \frac{9}{4}$

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