f'(c) = [f(b) - f(a)]/(b - a).
By breaking down the formula that defines the Mean Value Theorem
-Tangent line's slope at point c defined by f'(c) =
-Secant line slope defined by [f(b) - f(a)]/(b -a).
This means that in the interval [a,b] there will be a point c whose tangent line's slope will be the same as the slope from the secant line between points a and b, which will make both lines parallel to each other.
Graphically ( to understand it better):
I'm going to use f(x)= √x +2 as an example.
The red line it's f(x)= √x +2
The green line is the secant line ([f(b) - f(a)]/(b -a)) between the interval [1,25] where a=1 and b=25.Secant line equation= f(x)= 1/6x+2.84
Secant line slope= 1/6
Now according to the Mean Value Theorem there is going to be a point c in between this interval whose tangent line's slope will be the same as the slope of this green secant line.
In order to find that point c, we look for the derivative of f(x)= √x +2 which is the slope for f(x)= √x +2. The derivative is f'(x)= 1/2√x and we set it equals to the slope from the secant line, like this
1/6=1/2√x
and at the end we get x= 9
9 it's our point c where its tanget line will have the same slope as the secant line!
In this way, after plugging in x=9 in f(x)= √x +2 we get the coordinates (9, 5), which is the point for the tanget line.
The pink line it's the tangent line at point c which is (9,5) that has the same slope as the secant line in the interval [1,25] thus making them parallel to each other.
Tangent line equation= f(x)= 1/6x+3.5
Tangent slope= 1/6
2.- However, the Mean Value Theorem fails if in the interval [a,b] the function is discontinuous or not differentiable.
DISCONTINUOUS:
Here the Mean Value Theorem Fails because in the interval [a,b] the function is discontinuous and there is no guaranteed point c between [a,b] because the function has a gap and it doesn't satisfy the continuity requirement. (NOTE: L in this graph is just to prove the Intermediate Value Theorem, which is different from the Mean Value 
Theorem. I used the graph due to the discontinuity it shows.)However, I fixed the graph and this one to the right its clearer!!!
NOT DIFFERENTIABLE:
.The Mean Value Theorem fails in the interval [-4,4] because it has a corner on x=0 which makes the function not differentiable at that point. This means that a tangent line at x=0 doesn't exist.
Here the Mean Value Theorem Fails because in the interval [a,b] the function is discontinuous and there won't be any point c between [a,b] because it just doesn't exist.
The Mean Value Theorem fails in the interval [-4,4] because it has a corner on x=0 which makes the function not differentiable at that point. This means that a tangent line at x=0 doesn't exist. 