The above figure is the graph of a continuous and differentiable function y = f(x). Between point A & B the function has its derivative zero at how many points -
We are given the function is continuous and differentiable. We want to find the number of points where the derivative of f(x) becomes zero. Derivative becomes zero means the slope of the tangent also becomes zero at that point. We can guess by seeing the graph that there are 5 such places between A and B, corresponding to a maxima or minima.
We can understand this using Rolle’s theorem also. For that, let the points where the straight line cuts the graph between A and B be a1, a2, a3 and a4. Now, if we consider the interval [A, a1], we will find that Rolle’s theorem is applicable there. Function is continuous and differentiable as given in the question and the value of the function at A and a1 are equal, because the line cuts the graph at a1 and A. According to Rolle’s theorem, there should be at least one point in the interval [A, a1], where f’(x) becomes zero.
Since we can have 5 such intervals, there will be total 5 points where the derivative becomes zero.