First slide

Extreme values and Periodicity of Trigonometric functions

Question

The difference of the maximum and the minimum values of the function $5 \cos x + 3 \cos (x + \frac{π}{3}) + 8$ over R is

Moderate

A

16
B

14
C

7
D

15

Solution

$5 \cos x + 3 (\frac{\cos x}{2} - \frac{\sqrt{3}}{2} \sin x) + 8 = \frac{13}{2} \cos x - \frac{3 \sqrt{3}}{2} \sin x + 8$
Whose minimum and maximum values are
$\begin{array}{l} 8 - \sqrt{{(\frac{13}{2})}^{2} + {(\frac{3 \sqrt{3}}{2})}^{2}} a n d 8 + \sqrt{{(\frac{13}{2})}^{2} + {(\frac{3 \sqrt{3}}{2})}^{2}} \\ = 1 a n d 15 \end{array}$
$∴$ Maximum – minimum = 15 – 1 = 14

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