Edit of Theorem

1) The Mean Value Theorem states that on a continuous and differentiable function there is a point where the slope of the function is parallel to the tangent line.If the constant C is between the interval of [a,b] , there is at least one point when the secant line is parallel to the tangent line.

OMG is took me 30 minds to do the graph -_-

The graph shows that after much unwanted calculations for the secant and tangent lines, they appear to be parallel because their is a Constant on the curve of the function that makes the tangent line parallel to the secant line.

2) The Mean value theorem doesn't apply because at x=0 the slope of the tangent line is undefined. Thus there would be no point that is guaranteed to be a point C in the intervals [a,b].

God says that the mean value theorem only works for continuous functions , but this one is a discontinuous.. So god is right. I think.

MEAN value theorem

1. The mean value theorem means that the graph is continuous , differentiable and defined thoughout the function. The graph shows two parallel lines that are passing through points [a, f(a)] and [b , f(b)]. The theorem states that within [a,b] , there is a point where the slope of the parallel lines are the same.

The graph shows a jump discontinuity and a corner .

( I used Eric's example )

The function is on the interval of [0,7] and shows a discontinuous function. It is clearly shown that

points A to B has a slope that is not parallel to the rest of the function. Points C to D show a corner discontinuity.

This shows that when X=5 there is a corner. and the slope of the line between C to D is =0 , therefore Undefined. Since the slope of each

portion [a,b] and [c,d] are different slopes , the Mean Theorem cannot apply to a discontinuous function because their isnt a point where the slope of the differential function is equivalent to the derivative .