On which of the following intervals is the function f is given by f (x) = x¹⁰⁰ + sin x - 1 is strictly decreasing?
(A) (0, 1)   (B) (π/2, π)   (C) (0, π/2)   (D) None of these

Solution:

Increasing functions are those functions that increase monotonically within a particular domain,

and decreasing functions are those which decrease monotonically within a particular domain.

We have,

f (x) = x¹⁰⁰ + sin x - 1

Therefore,

f' (x) = 100x⁹⁹ + cos x

In interval (0, 1),

cos x > 0 and 100x⁹⁹ > 0

Hence, f' (x) > 0

Thus, f is strictly increasing in (0, 1)

In interval (π/2, π),

cos x < 0 and 100x⁹⁹ > 0

Hence, f' (x) > 0

Thus, f is strictly increasing in interval (π/2, π)

Now, in interval (0, π/2)

cos x > 0 and 100x⁹⁹ > 0

Hence, f' (x) > 0

Thus, f is strictly increasing in the interval (0, π/2)

Hence, f is strictly decreasing in none of the intervals.

Thus, the correct option is D

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 13

On which of the following intervals is the function f is given by f (x) = x¹⁰⁰ + sin x - 1 is strictly decreasing? (A) (0, 1) (B) (π/2, π) (C) (0, π/2) (D) None of these.

Summary:

For the function given as f (x) = x¹⁰⁰ + sin x - 1.we have observed that f is strictly increasing in (0, 1), f is strictly increasing in the interval (π/2, π) f is strictly increasing in the interval (0, π/2). Hence, f is strictly decreasing in none of the intervals. Thus, the correct option is D