The Mean Value Theorem

1. The mean value theorem states that when given a function f(x), the slope of the secant line between two points f(b)-f(a) is equal to the slope of the tangent line at c

For example:
When f(x)=x² from [-4,4]
you find that the secant line is y=2x by using (f(b)-f(a))/(b-a) where b=2 and a=-2

in order to find the tangent line, you have to take the derivative of the original equation which would be f(x)=x²
which is equal to 2x this means that f(c)=2c
you set that equal to the slope which is 2 to get 2c=2
then you get c=1
now you imput that into the original equation
f(1)=1²=1
1 is the output or the y so the tangent is at point (1,1)

Now to find the equation for the tangent
y-y1=m(x-x1)
y-1=2(x-1)
y-1=2x-2
y=2x-1

and theres the equation

as you can see in the graph, both the tangent and the secant line are parallel

2. When a function is discontinuous or not differentiable, the mean value theorem will fail. there may be a secant line but there will not be a tangent parallel to it. For example if the graph is an absolute value graph, the secant line and the tangent lines will not be parallel. The slope of the secant would be 0. The only place where there could be a parallel tangent is at x=0 but there isn't because at x=0 it is non differentiable meaning there is no slope at that point. If the graph is discontinuous, the mean value theorem fails because there is no guaranteed point at x=c, meaning that not always will there be a tangent that is parallel to the secant line, because the point where the tangent line should be might not exist. There might be an equation like f(x)=1/x², in this function, there is no output at x=0. The slope of the secant line is 0 and there is no point in the function in which the slope of the tangent line can be 0 since there is no output at x=0