Question

Verify Lagrange's Mean Value Theorem for the following function:

`f(x ) = 2 sin x +  sin 2x " on " [0, pi]`

Answer 1

`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`