Show that the function given by f (x) = sin x is
(a) f (x) = sin x is (0, π/2)
(b) Strictly decreasing in (π/2, π)
(c) Neither increasing nor decreasing in (0, π)

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.

It is given that f (x) = sin x

Hence, f' (x) = cos x

(a) Here, x ∈ (0, π/2)

⇒ cos x > 0

⇒ f' (x) > 0

If the derivative is greater than 0 then the function is an increasing function.

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

(b) Here, x ∈ (π/2, π)

⇒ cos x < 0

⇒ f' (x) < 0

If the derivative is lesser than 0 then the function is a decreasing function. 

Thus, f is strictly decreasing in (π/2, π)

(c) Here, x ∈ (0, π)

The results obtained in (a) and (b) are sufficient to state that f is neither increasing nor decreasing in (0, π)

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 3

Show that the function given by f (x) = sin x is (a) f (x) = sin x is (0, π/2) (b) Strictly decreasing in (π/2, π) (c) Neither increasing nor decreasing in (0, π)

Summary:

(a) f is strictly increasing in (0, π/2) (b) f is strictly decreasing in (π/2, π) (c) The results obtained in (a) and (b) are sufficient to state that f is neither increasing nor decreasing in (0, π)