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Question: The range of \(f(x) = \sec\left( \frac{\pi}{4}\cos^{2}x \right), - \infty < x < \infty\) is...

The range of f(x)=sec⁡(π4cos⁡2x),−∞<x<∞f(x) = \sec\left( \frac{\pi}{4}\cos^{2}x \right), - \infty < x < \infty is

A

[1,2]\lbrack 1,\sqrt{2}\rbrack

B

[1,∞)\lbrack 1,\infty)

C

[−2,−1]∪[1,2]\lbrack - \sqrt{2}, - 1\rbrack \cup \lbrack 1,\sqrt{2}\rbrack

D

(−∞,−1]∪[1,∞)( - \infty, - 1\rbrack \cup \lbrack 1,\infty)

Answer

[1,2]\lbrack 1,\sqrt{2}\rbrack

Explanation

Solution

f(x)=sec⁡(π4cos⁡2x)f(x) = \sec\left( \frac{\pi}{4}\cos^{2}x \right)

We know that, 0≤cos⁡2x≤10 \leq \cos^{2}x \leq 1 at cos⁡x=0,f(x)=1\cos x = 0,f(x) = 1

and at cos⁡x=1,\cos x = 1, f(x)=2f(x) = \sqrt{2}

∴ 1≤x≤21 \leq x \leq \sqrt{2} ⇒ x∈[1,2]x \in \lbrack 1,\sqrt{2}\rbrack.