# Applications of the Mean Value Theorem

What are some interesting applications of the Mean Value Theorem for derivatives (both the 'extended' or 'non-extended' versions as seen here are of interest)?

So far I've seen some trivial applications like finding the number of roots of a polynomial equation. What are some more interesting applications of it?

I'm asking this as I'm not exactly sure why MVT is so important - so examples which focus on explaining that would be appreciated.

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The fundamental theorem of calculus is already enough of a reason to teach people the MVT, I think. –  Qiaochu Yuan Nov 10 '10 at 20:32
An interesting discussion of the MVT on Timothy Gowers's web page: dpmms.cam.ac.uk/~wtg10/meanvalue.html –  Hans Lundmark Nov 12 '10 at 8:11

There are several applications of the Mean Value Theorem. It is one of the most important theorems in analysis and is used all the time. I've listed $5$ important results below. I'll provide some motivation to their importance if you request.

$1)$ If $f: (a,b) \rightarrow \mathbb{R}$ is differentiable and $f'(x) = 0$ for all $x \in (a,b)$, then $f$ is constant.

$2)$ Leibniz's rule: Suppose $f : [a,b] \times [c,d] \rightarrow \mathbb{R}$ is a continuous function with $\partial f/ \partial x$ continuous. Then the function $F(x) = \int_{c}^d f(x,y)dy$ is derivable with derivative $$F'(x) = \int_{c}^d \frac{\partial f}{\partial x} (x,y)dy.$$

$3)$ L'Hospital's rule

$4)$ If $A$ is an open set in $\mathbb{R}^n$ and $f:A \rightarrow \mathbb{R}^m$ is a function with continuous partial derivatives, then $f$ is differentiable.

$5)$ Symmetry of second derivatives: If $A$ is an open set in $\mathbb{R}^n$ and $f:A \rightarrow \mathbb{R}$ is a function of class $C^2$, then for each $a \in A$, $$\frac{\partial^2 f}{\partial x_i \partial x_j} (a) = \frac{\partial^2 f}{\partial x_j \partial x_i} (a)$$

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There are applications.

For an important one, Taylor series proof relies on it.

An other application I like is to quickly come up with and prove inequalities.

Example 1) $\displaystyle |\cos x - \cos y| \le |x - y|$

Example 2) $\displaystyle \frac{1}{2\sqrt{n+1}} < \sqrt{n+1} - \sqrt{n} < \frac{1}{2\sqrt{n}}$

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Some more applications:

• If the derivative of a function $f$ is everywhere strictly positive, then f is a strictly increasing function.

• Suppose $f$ is differentiable on whole of $\mathbb{R}$, and $f'(x)$ is a constant. Then $f$ is linear.

• Mean Value theorem plays an important role in the proof of Fundamental Theorem of Calculus.

• Suppose $f$ is continuous on $[a,b]$ and $f'$ exists and is bounded on the interior, then $f$ is of Bounded Variation on $[a,b]$.

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The generalization of the Mean Value Theorem known as Cauchy's Mean Value Theorem can be used to prove L'Hopital's Rule. See for instance the following proof.

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If $f$ and $g$ are two differentiable functions on an interval $I$ and for all $x\in I$, $f^{\prime }(x)=g^{\prime }(x)$, then the difference $f(x)-g(x)$ is constant on $I$.

Physical interpretation (assuming the movement is uninterruptible): If the average speed of a car between two locations was $V$ km/h, then there was at least one instant where the speed indicator displayed $V$ km/h.

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