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.
Question to think about. 1. Is the given function satisfying all conditions of Rolle’s theorem? 2. Is it possible to get f’(x)≠0 at a point between points A and B or B and C?