Rolle's theorem

On the screen, we can see a graph of the function f(x)=ax^3+bx^2+cx+d, where we can change the values of a, b, c and d using their corresponding slide bars. As this function is a polynomial so it is continuous and differentiable for all values of x. We can also see three points A, B and C on this graph and their y coordinates are equal to 5. We know from Rolle’s theorem that if a function is continuous and differentiable and f(a)=f(b) then there will be a point in between (a, b) where f’(x)=0 so let us move point D using the mouse and observe how m gets changed. m = slope of the tangent at point D. We know that slope of the tangent represents f’(x) at that point D.