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

The Correct Option is D

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).