Geometric Interpretation of Rolle’s Theorem

We can visualize Rolle’s theorem from the figure(1) added below,

In the above figure the function satisfies all three conditions given above. So, we can apply Rolle’s theorem, according to which there exists at least one point ‘c’ such that:

f'(c) = 0

which means that there exists a point at which the slope of the tangent at that is equal to 0. We can easily see that at point ‘c’ slope is 0. Similarly, there could be more than one points at which slope of tangent at those points will be 0. Figure(2) added below is one of the example where exists more than one point satisfying Rolle’s theorem.

Rolle’s theorem one of the core theorem of calculus states that, for a differentiable function that attains equal values at two distinct points then it must have at least one fixed point somewhere between them where the first derivative of the function is zero.