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

The derivative of a differentiable function vanishes at an extremum(ie maximum or minimum) point.

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    $\begingroup$ This does not need the Mean Value Theorem, but only the limit definition of the derivative. $\endgroup$ – KCd Oct 22 '15 at 12:37
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MVT is very important. In calculus and analysis, of course. But it's important in other areas too, like applied mathematics an even number theory. For example, for showing Liouville numbers are transcendental.

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