Applications of the Mean Value Theorem

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 $a \in A$, $$\frac{\partial^2 f}{\partial x_i \partial x_j} (a) = \frac{\partial^2 f}{\partial x_j \partial x_i} (a)$$

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

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 $[a,b]$.