First slide

Extreme values of trigonometric functions in trigonometry

Question

The maximum value of $4 \sin^{2} x + 3 \cos^{2} x + \sin \frac{x}{2} + \cos \frac{x}{2} is$

Moderate

A

$4 + \sqrt{2}$
B

$3 + \sqrt{2}$
C

9
D

4

Solution

Maximum value of $4 \sin^{2} x + 3 \cos^{2} x i.e., \sin^{2} x + 3 is 4$ and that of sin $\frac{x}{2} + \cos \frac{x}{2} is \frac{1}{\sqrt{2}} + \frac{1}{\sqrt{2}} = \sqrt{2}$ ,both attained $at x = \frac{π}{2} .$ Hence, the given function has maximum value $4 + \sqrt{2}$

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