Q. 65.0( 5 Votes )

Find the value of

Answer :

Given: f(x) = x- 3x

Necessary conditions for rolle's theorem:

1. f(x) is continuous on [a, b].
2. f(x) is differentiable on (a, b).
3. f(a) = f(b)

Explanation:


Since a polynomial function is everywhere continuous and differentiable, therefore f(x) is continuous on [-√3, 0] and differentiable on [-√3, 0].

Also, f(-√3) = (-√3)3 - 3(-√3) = -3√3 + 3√3 = 0


and, f(0) = (0)3 - 3(0) = 0


f(-√3) = f(0)


Thus, all the three conditions of Rolle's theorem are satisfied.


Thus, there exists c (-√3, 0) such that f'(c) = 0


Now, f'(x) = 3x2 - 3


f'(x) = 0


3x2 - 3 = 0


3(x2 - 1) = 0


x2 - 1 = 0


x = ± 1


Clearly, only -1 lies in the interval (-√3, 0).


Thus, the value of c is -1.

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