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Question
Verify Lagrange's Mean Value Theorem for the following function:
`f(x ) = 2 sin x + sin 2x " on " [0, pi]`
Solution
`f(x) = 2 sin x + sin 2x " on " [0, pi]`
`f'(x) = 2cosx + 2cos 2x`
1) f(x) is differentiable on `[0, pi]`
2) Differentibility ⇒ Continuity
:. f(x) is continuous on `[0, pi]`
∴ LMVT is verified
then there exist `c in (0,pi)` such that
`f'(c) = (f(b) - f(a))/(b-a)`
`2cos c + 2 cos c = ((2sin pi + sin 2pi) - (2sin 0 +sin 0)) /(pi-0)`
`2 cos c + 2cos2c = 0`
`2cos c + 2(2cos^2 c - 1) = 0`
`2cos^2c + 2cos c -1 = 0`
`2 cos^2 c + 2 cos c - cos c - 1 = 0`
`2cos(cos c + 1) -1(cos c + 1) = 0`
`(cos c + 1)(2cos c - 1) = 0`
`cos c = -1, cos c = 1/2`
`c = 0 ∉ (0, pi)`
`c= pi/3 in (0, pi`)`
`:. c =pi/3`
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