Question

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

• A. $(0,1)$
• B. $(\frac \pi2,\pi)$
• C. $(0,\frac \pi2)$
• D. $None\ of \ these$

Answer 1

Answer: (D)

$None\ of \ these$

Answer 2

We have,

f(x) = x100+sinx-1

f'(x) = 100x99+cosx

In interval(0,1), cosx>0 and 100x99>0.

f'(x)>0.

Thus, function f is strictly increasing in interval (0, 1).

In interval ($\frac \pi2$,$\pi$), cosx<0 and 100x99>0. also, 100x100>cosx

$\implies$f'(x)>0 in ($\frac \pi2$,$\pi$).

Thus, function f is strictly increasing in interval ($\frac \pi2$,$\pi$).

In interval (0,$\frac \pi2$), cosx>0 and 100x99>0.

100x99+cosx>0

f'(x)>0 on (0,$\frac \pi2$).

Hence, function f is strictly decreasing in none of the intervals. The correct answer is (D).