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If $f (x) = 2 + \cos^{2} x + \sec^{2} x,$ its value always is

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$f (x) < 1$

$f (x) = 1$

$f (x) \geq 4$

$2 > f (x) > 1$

answer is C.

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Detailed Solution

$f (x) = 2 + \cos^{2} x + \sec^{2} x$

$= 2 + \cos^{2} x + \frac{1}{\cos^{2} x} \geq 2 + 2 (∵ A . M \geq G . M \Rightarrow \cos^{2} x + \frac{1}{\cos^{2} x} \geq 2 \sqrt{\cos^{2} x \frac{1}{\cos^{2} x}} = 2)$

$\Rightarrow f (x) \geq 4$

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