Q. The minimum value of $f(x) = \sin^{4}{x + \cos^{4}x},\ 0 \leq x \leq \frac{\pi}{2}$ is
A) $\frac{1}{2\sqrt{2}}$
B) $\frac{1}{4}$
C) $\frac{- 1}{2}$
D) $\frac{1}{2}$
Explanation
(d)
Given, $f(x) = \sin^{4}{x + \cos^{4}x}$
$= \left( \sin^{2}{x + \cos^{2}x} \right)^{2} - 2\sin^{2}x\cos^{2}x$
$\Rightarrow \ \ \ f(x) = 1 - \frac{1}{2}\sin^{2}{2x}$
Also, $0 \leq \sin^{2}{2x} \leq 1$
$\therefore$ Minimum value of $f(x)$ is $1 - \frac{1}{2} = \frac{1}{2}$
Reviewed by Dr. Abubacker
Physics Educator & NEET/JEE Specialist
Ph.D. in Physics with 19+ years of teaching experience.