Prove that the function f given by f (x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, 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.

Logs (or) logarithms are nothing but another way of expressing exponents.

We have

f(x) = log sin x

Therefore,

f'(x) = 1/sin x (cos x)

= cot x

In the interval (0, π/2),

f'(x) = cot x > 0

Hence,

f is strictly increasing in (0, π/2)

In the interval (π/2, π), f' (x)

= cot x < 0

Hence,

f is strictly decreasing in (π/2, π)

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NCERT Solutions Class 12 Maths - Chapter 6 Exercise 6.2 Question 16

Prove that the function f given by f (x) = log sin x is increasing on (0, π/2) and decreasing on (π/2, π)

Summary:

Hence we have proved that the function f given by f (x) = log sin x is strictly increasing on (0, π/2) and strictly decreasing on (π/2, π)