Question

The largest interval lying in $\left(\frac{-\pi}{2}, \frac{\pi}{2}\right)$ for which the function $f(x)=4^{-x^2}+\cos ^{-1}\left(\frac{x}{2}-1\right)+\log (\cos x)$ is defined, is

(1) $[0, \pi]$

(2) $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$

(3) $\left[-\frac{\pi}{4}, \frac{\pi}{2}\right)$

(4) $\left[0, \frac{\pi}{2}\right)$

Answer 1

SOLUTION :  $f(x)$ is defined if $-1 \leq \frac{x}{2}-1 \leq 1$ and $\cos x>0$

or $\quad 0 \leq \frac{x}{2} \leq 2$ and $-\frac{\pi}{2}<x<\frac{\pi}{2} \quad$ or $\quad 0 \leq x \leq 4$ and $-\frac{\pi}{2}<x<\frac{\pi}{2} \quad \therefore \quad x \in\left[0, \frac{\pi}{2}\right)$. 

Hence Option 4 is Correct.