Rolle's theorem in Calculus explained

Rolle's theorem is slightly different from Mean Value Theorem explained as simple as possible

1. If a real valued function f(X) is continuous over a closed interval [a,b]
2. Function is differentiable over an open interval (a,b)
3. And if, f(a) = f(b) (which means if we substitute values 'a' and 'b' to the given function f(x), the corresponding value of 'y' will be same at 'a' and 'b' and obviously the line joining the endpoints will have ' zero' slope or in short, this line will be parallel to 'X' axis)

Then there exists at least one point 'C' such that derivative OR dy/dx=0 at that point 'C'  or in simple words if we draw a tangent to the curve at point 'C' that tangent will have zero slope or tangent will be parallel to 'X' axis.

Ex: f(x) = x2+2  over [-1, 1]  since this function is a polynomial,  function is continuous between [-1, 1] and differentiable over the interval (-1,1). Here f(-1) = 3 and f(1) = 3 ie: value of 'y' at x= (-1) and value of 'y' at x= (1) are equal. For this function if we join the endpoints of the curve from x = -1 to x = (1) , this line will have 'zero' slope. Now if you differentiate this function wrt to 'X'    y'= 2X. At what value of 'X'   y' =o ?  Yes !!  It is at 'X' = 0. So for this function the point 'C' is at 'X'= 0. Means if we draw a tangent to this curve at x=0, tangent will have zero slope or tangent is parallel to 'X' axis.