First slide

Extreme values and Periodicity of Trigonometric functions

Question

Let $f (x) = \cos x (\sin x + \sqrt{\sin^{2} x + \sin^{2} θ})$ where $θ$ is a given constant, then maximum value of $f (x)$ is

Moderate

A

$\sqrt{1 + \cos^{2} θ}$
B

$| \cos θ |$
C

$\sqrt{1 + \sin^{2} θ}$
D

$| \sin θ |$

Solution

$f (x) = \cos x (\sin x + \sqrt{\sin^{2} + \sin^{2} θ})$
$\Rightarrow {(f (x) \sec x - \sin x)}^{2} = \sin^{2} x + \sin^{2} θ$
$\Rightarrow f^{2} (x) \sec^{2} x - 2 f (x) \tan x = \sin^{2} θ$
$\Rightarrow f^{2} (x) \tan^{2} x - 2 f (x) \tan x +$ $f^{2} (x) - \sin^{2} θ = 0$
$∴ 4 f^{2} (x) \geq 4 f^{2} (x) (f^{2} (x) - \sin^{2} θ)$
$\Rightarrow f^{2} (x) \leq 1 + \sin^{2} θ \Rightarrow | f (x) \leq \sqrt{1 + \sin^{2} θ} |$

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