Question

Verify Mean value theorem for the function f(x) = 2sin x + sin 2x on [0, π].

Answer 1

f(x) = 2sin x + sin 2x on [0, π].
f(x) is continuous in [0, π].
f(x) is differentiable in [0, π].

∴ Mean value theorem is applieable
f(0) = 0, f(x) = 0
f'(x) = 2 cos x + 2 cos 2x
f'(c) = 2 cos c + 2 cos 2c

f'(c) = `(f(pi) - f(0))/(pi - 0) = 0`

∴ 2 cos c + 2 cos 2c = 0
⇒ (2 cos c - 1)(cos c + 1) = 0

⇒ cos c`1/2`

⇒ c = `pi/3` ∈ (0, π)

Hence mean value theorem is verified.