1. This means that if you take a function, f(x)
then the slope of the secant line between two points, [f(b)-f(a)]/(b-a)
is equal to the slope of the tangent line at the point c, f'(c)
So for example:
f(x)=x²+1 (to make things easier to see)
f'(x)=2x
[f(b)-f(a)]/(b-a) where a=-2 & b=2 ==> [4-4]/(2+2)=0
f'(c) = [f(b)-f(a)]/(b-a) ==> 2c=0 ==> c=0
Basically, it means the the secant line and the tangent line are parallel to each other.
2. This only works for continuous and differentiable functions because if the function is neither, then either the point c or the slope at c may not exist, as in the function
f(x)=|x| where a=-2 and b=2 (sorry, no graph this time). In this case, the point c would have to be at 0, but since the function is not differentiable at 0, f'(c) does not exist and the mean value theorem does not apply.
