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