Mean Value Theorem

This illustration of the Mean Value Theorem with an optional point that is not differentiable. The curve can be modified by moving the black points. The Mean Value Theorem states If is continuous on the interval and differentiable on the interval then there exist at least one point, , in the interval such that Checking Rolle's Theorem will modify the function to make the end points have equal values. Rolle's Theorem states If is continuous on the interval , differentiable on the interval and then there exist at least one point, , in the interval such that Orange X symbols show where these points are in the interval. The Tangents check box will show the tangent line through the points that satisfy the derivative constraint. The Ends Slope will show the line through the end points with the slope for both theorems.
Can you adjust the curve and boundary points so that there are no X points shown? What is true when no X point is shown? Can more than one point satisfy the derivative value? Can you explain the movement of the X points for the Mean Value Theorem?