Question

If f(x) = log (sin x), x ∈ `[π/6, (5π)/6]`, then value of 'c' by applying LMVT is ____________.

Options

• `π/2`

• `(2π)/3`

• `(3π)/6`

• `π/4`

Answer 1

If f(x) = log (sin x), x ∈ `[π/6, (5π)/6]`, then value of 'c' by applying LMVT is `underline(π/2)`.

Explanation:

Given, f(x) = log (sin x)

a = `π/6`, b = `(5π)/6`

By Lagrange's mean value theorem (L.M.V.T)

f'(c) = `("f"("b") - "f"("a"))/("b" - "a")`

∴ f'(x) = cot x

f(b) = `log (sin  π/6) = log (1/2) = - log 2`

f(a) = `log (sin  (5π)/6) = log (1/2) = - log 2`

∴ f'(c) = cot C = `(- log 2 + log 2)/((5π)/6 - π/(π/6))` = 0

= cot C = 0 = C = `π/2`