Q. 65.0( 5 Votes )

# Find the value of

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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