Minimum value of the function f(x) = | ,x + ,x + ,x + ,x + ,x + c… - Vidyadip

MCQ

Minimum value of the function $f(x) = \left| {\sin \,x + \cos \,x + \tan \,x + \cot \,x + \sec \,x + \ cosec\ x} \right|$ is equal to

A
$2\sqrt 2$
✓
$2\sqrt 2 - 1$
C
$2 + 3\sqrt 2 $
D
$2\sqrt 2 + 1$

✓

Answer

Correct option: B.

$2\sqrt 2 - 1$

b
Put $=\sin x+\cos x=p \Rightarrow 1+2 \sin x \cos x=p^{2}$

$\sin x \cos x=\left(\frac{p^{2}-1}{2}\right)$

$\therefore f(x) = \left| {p + \frac{2}{{\left( {{p^2} - 1} \right)}} + \frac{{2p}}{{{p^2} - 1}}} \right|$

${=\left|p+\frac{2}{(p-1)}\right|} $

${=\left|(p-1)+\frac{2}{(p-1)}+1\right|}$

$\therefore \frac{(p-1)+\left(\frac{2}{p-1}\right)}{2} \geq\left((p-1) \cdot \frac{2}{p-1}\right)^{1 / 2}$

$\therefore(p-1)+\left(\frac{2}{p-1}\right) \geq 2 \sqrt{2}$

$\therefore \quad f(x)_{\min } \geq 2 \sqrt{2}+1$

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