Q. 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
A) $\lbrack 0,\ \pi\rbrack$
B) $\left( - \frac{\pi}{2},\ \frac{\pi}{2} \right)$
C) [image:2_image2.emf]
D) $\left\lbrack 0,\ \frac{\pi}{2} \right\rbrack$
Explanation
(d)
Given, $f(x) = 4^{- x^{2}} + \cos^{- 1}{\left( \frac{x}{2} - 1 \right) + \log\left( \cos x \right)}$
Here, $4^{- x^{2}}$ is defined for $\left\{ - \frac{\pi}{2},\ \frac{\pi}{2} \right\},\cos^{- 1}\left( \frac{x}{2} - 1 \right)$ is defined,
If $- 1 \leq \frac{x}{2} - 1 \leq 1 \Rightarrow 0 \leq x \leq 4$
And $\log{(\cos x)}$ is defined, if $\cos{x > 0}$
$\Rightarrow \ - \frac{\pi}{2} < x < \frac{\pi}{2}$
Hence, $f(x)$ is defined for $x \in \left\lbrack 0,\ \frac{\pi}{2} \right\rbrack$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.