Intuition Behind the Mean Value Theorem
The mean value theorem is one of the "big" theorems in calculus. In this page I'll try to give you the intuition and we'll try to prove it using a very simple method.
This theorem is very simple and intuitive, yet it can be mindblowing. Suppose you're riding your new Ferrari and I'm a traffic officer. I suspect you may be abusing your car's power just a little bit.
I know you're going to cross a bridge, where the speed limit is 80km/h (about 50 mph). So, I just install two radars, one at the start and the other at the end. The first one will start a chronometer, and the second one will stop it.
I also know that the bridge is 200m long. So, suppose I get:
Total crossing time: 8 seconds.
Your average speed is just total distance over time:
So, your average speed surpasses the limit. Does this mean I can fine you? Unfortunatelly for you, I can use the Mean Value Theorem, which says:
"At some instant you where actually travelling at the average speed of 90km/h".
The Theorem
In terms of functions, the mean value theorem says that given a continuous function in an interval [a,b]:
There is some point c between a and b, that is:
Such that:
That is, the derivative at that point equals the "average slope". What is the right side of that equation? Let's look at it graphically:
The expression 
is the slope of the line crossing the two endpoints of our function.
If the function represented speed, we would have average speed: change of distance over change in time. The derivative f'(c) would be the instantaneous speed.
There is also a geometric interpretation of this theorem. We just need to remind ourselves what is the derivative, geometrically: the slope of the tangent line at that point.
So, the mean value theorem says that there is a point c between a and b such that:
The tangent line at point c is parallel to the secant line crossing the points (a, f(a)) and (b, f(b)):
Preparing for the Proof: Rolle's Theorem
The proof of the mean value theorem is very simple and intuitive. We just need our intuition and a little of algebra. To prove it, we'll use a new theorem of its own: Rolle's Theorem.
This theorem says that given a continuous function g on an interval [a,b], such that g(a)=g(b), then there is some c, such that:
And:
Graphically, this theorem says the following:
Given a function that looks like that, there is a point c, such that the derivative is zero at that point. That implies that the tangent line at that point is horizontal. Why? Because the derivative is the slope of the tangent line. Slope zero implies horizontal line.
This one is easy to prove. We know that the function, because it is continuous, must reach a maximum and a minimum in that closed interval. Let's call:
M: maximum of g(x) m: minimum of g(x)
We have just two possibilities:
- M = m
- M > m
If M = m, we'll have that the function is constant, because f(x) = M = m. So, f'(x) = 0 for all x. And we not only have one point "c", but infinite points where the derivative is zero.
If M > m, we have again two possibilities:
- M = f(a)
- M is distinct from f(a)
If M = f(a), we also know that f(a)=f(b), so, that means that f(b)=M also. That in turn implies that the minimum m must be reached in a point between a and b, because it can't occur neither in a or b.
If M is distinct from f(a), we also have that M is distinct from f(b), so, the maximum must be reached in a point between a and b. And as we already know, in the point where a maximum or minimum ocurs, the derivative is zero.
The Proof
Now, the mean value theorem is just an extension of Rolle's theorem. We just need a function that satisfies Rolle's theorem hypothesis. So, let's consider the function:
The function evaluated at "a" is:
Now, doing some algebra:
Now, let's do the same for the function g evaluated at "b":
We have that g(a)=g(b), just as we wanted. So, we can apply Rolle's Theorem now. What does it say? That there is a point c between a and b such that:
But what is g'(x)? It is this:
So, we finally have:
And that means that:
And that proves the Mean Value Theorem.