1. Rolle’s Theorem

2. Rolle's theorem essentially states that any real-valued differentiable function that attains equal values at two distinct points must have a stationary point somewhere between them; that is, a point where the first derivative

4. Q. Verify Rolle’s Theorem for the function f(x)=|x| ,-1 to +1

5. If differentiability fails at an interior point of the interval, the conclusion of Rolle's theorem may not hold.
Consider the absolute value function
Then f(−1) = f(1), but there is no c between −1 and 1 for which the derivative is zero.
This is because that function, although continuous, is not differentiable at x = 0.

6. The derivative of f changes its sign at x = 0, but without attaining the value 0.
The theorem cannot be applied to this function, clearly, because it does not satisfy the condition that the function must be differentiable for every x in the open interval.
However, when the differentiability requirement is dropped from Rolle's theorem, f will still have a critical number in the on interval (a,b).

7. Thank you