On which of the following intervals is the function $f$ given by $f(x) = {x^{100}} + \sin x - 1$ strictly decreasing?

(A) $\left( {0,1} \right)$

(B) $\left( {\cfrac{\pi }{2},\pi } \right)$

(C) $\left( {0,\cfrac{\pi }{2}} \right)$

(D) None of these

(D) We have, $f(x) = {x^{100}} + \sin x - 1$ …(i)
Differentiating (i) w.r.t. $x$, we get
$f'(x) = 100{x^{99}} + \cos x$

(A $f'(x)$ assumes only $+ ve$ values in $(0,1)$

Therefore, $f$ is strictly increasing in $(0,1)$

(B) $f'(x) > 0$ for all $x \in \left( {\cfrac{\pi }{2},\pi } \right)$, therefore $f$ is strictly increasing in $x \in \left( {\cfrac{\pi }{2},\pi } \right)$

(C) $f'(x) > 0$ for all $x \in \left( {0,\cfrac{\pi }{2}} \right)$, therefore $f$ is strictly increasing in $\left( {0,\cfrac{\pi }{2}} \right)$.