Mean Value Theorem

-

The mean value theorem (MVT) says that, for a given arc connecting two points of a function, there is at least one point at which the slope of the tangent line is equal to the slope of the arc.



Theorem (Rolle's theorem)

Let \(f(x)\) be a continuous function defined on a closed interval \([a, b]\). Suppose \(f(x)\) is differentiable on the open interval \((a, b)\) and \(f(a) = f(b)\). Then there exists a \(c\in (a,b)\) such that \(f'(c) = 0\).

Proof. If the (global) maximum and minimum are both at \(x=a\) and \(x=b\), then \(f(x)\) is constant since \(f(a) = f(b)\) (= maximum = minimum). In this case, \(f'(x)=0\) at all \(x \in (a,b)\). Otherwise, \(f(x)\) has a maximum or minimum value at some \(c \in (a,b)\) so that \(f'(c) = 0\). ■

Theorem (Mean value theorem)

Let \(f(x)\) be a continuous function on \([a,b]\) such that it is differentiable on \((a,b)\). Then there exists a \(c \in (a,b)\) such that

\[f'(c) = \frac{f(b) - f(a)}{b - a}.\]

Proof. Let us define

\[g(x) = f(x) - \frac{f(b) - f(a)}{b - a}\cdot(x - a).\]

Then, \(g(x)\) is continuous on \([a, b]\) and differentiable on \((a,b)\), and \(g(a) = g(b) = f(a)\). Hence, by Rolle's theorem, there exists a \(c\in(a,b)\) such that \(g'(c) = 0\). But

\[g'(c) = f'(c) - \frac{f(b) - f(a)}{b - a} = 0\]

so

\[f'(c) = \frac{f(b) - f(a)}{b - a}.\]

■

Let's see some applications of the mean value theorem.

Corollary

Suppose that the function \(f(x)\) is differentiable on the open interval \(I=(a,b)\) and identically \(f'(x) = 0\) on \(I\). Then \(f(x)\) is a constant function on \(I\). That is, \(f(x) = C\) for all \(x \in I\) where \(C\) is a constant.

Proof. Let us pick an arbitrary \(c\in (a,b)\) and let \(C = f(c)\). We show that for all \(x\in(a,b)\), \(f(x) = C\). If \(x=c\), then \(f(x) = C\) by definition of \(C\). Suppose \(x\neq c\). If \(x < c\), consider the closed interval \([x, c]\). Since \(f(x)\) is differentiable on \((a,b)\), it is continuous on \([x, c]\). By the mean value theorem, there is a \(d\in(x,c)\) such that \(f(x) - f(c) = f'(d)(x-c)\). But since \(f'(d) = 0\) (by assumption), \(f(x) = f(c) = C\). If \(x > c\), we consider the closed interval \([c, x]\) and apply a similar argument. ■

Corollary (cor:monotone)

Suppose the function \(f(x)\) is continuous on the closed interval \([a,b]\) and differentiable on the open interval \((a,b)\).

  1. If \(f'(x) > 0\) for all \(x\in (a,b)\), then \(f(x)\) is strictly monotone increasing on \([a,b]\).
  2. If \(f'(x) < 0\) for all \(x\in (a,b)\), then \(f(x)\) is strictly monotone decreasing on \([a,b]\).
Proof. We only show part 1. Part 2 is similar.
It suffices to show that, for any \(c, d\in [a,b]\), if \(c < d\) then \(f(c) < f(d)\).
By assumption, \(f(x)\) is continuous on \([c,d]\) and differentiable on \((c,d)\). By the mean value theorem, there exists a \(t\in(c,d)\) such that \(f(d) - f(c) = f'(t)(d - c)\). But since \(f'(t) > 0\), we have \(f'(t)(d - c) > 0\) so that \(f(d) - f(c) > 0\). ■

Corollary (cor:minmax)

Suppose that the function \(f(x)\) is twice differentiable and that \(f'(c) = 0\) at some \(c\in (a,b)\).
  1. If \(f''(c) > 0\), then \(x = c\) is a local minimum point.
  2. If \(f''(c) < 0\), then \(x = c\) is a local maximum point.
Proof. We prove only part 1 (Part 2 is similar).
By assumption,
\[\lim_{x\to c}\frac{f'(x) - f'(c)}{x-c} = \lim_{x\to c}\frac{f'(x)}{x - c} = f''(c) > 0.\]
Let \(\varepsilon = f''(c) (> 0)\). There exists a \(\delta > 0\) such that, for all \(x \in (a,b)\),  \(0 < |x - c| < \delta\) implies \(\left|\frac{f'(x)}{x - c} - f''(c)\right| < \varepsilon\) or
\[-\varepsilon < \frac{f'(x)}{x - c} - f''(c) < \varepsilon.\]
It follows that
\[\frac{f'(x)}{x - c} > f''(c) -\varepsilon = 0.\]
In particular, if \(c - \delta < x < c\), then \(x - c < 0\) so \(f'(x) < 0\). That is, for all \(x \in (c -\delta, c)\), \(f(x) > f(c)\) (Corollary {cor:monotone}).

If \(c < x < c + \delta\), we have \(f'(x) > 0\) so, by the same argument (Corollary {cor:monotone}), \(f(c) < f(x)\).

In summary, for all \(x \in (c - \delta, c + \delta)\), if \(x \neq c\), then \(f(x) > f(c)\), which shows that \(f(c)\) is a local minimum value. ■

Definition (Convex, concave)

Let \(f(x)\) be a function defined on an open interval \(I\). \(f(x)\) is said to be convex at \(x= c\) if the graph of \(y= f(x)\) is above its tangent line at \(x=c\) in the neighbor of \(x=c\), that is, if there exists a \(\delta > 0\) such that, for all \(x\in I\),  \(0 < |x - c| < \delta\) implies \(f(x) > f'(c)(x - c) + f(c)\).

Similarly, \(f(x)\) is said to be concave at \(x = c\) if the graph of \(y = f(x)\) is below its tangent line at \(x=c\) in the neighbor of \(x=c\), that is, there exists a \(\delta > 0\) such that, for all \(x \in I\), \(0 < |x - c| < \delta\) implies \(f(x) < f'(c)(x - c) + f(c)\).

Remark. Recall that \(y = f'(c)(x - c) + f(c)\) is the equation of the tangent line of \(f(x)\) at \(x=c\). □

Corollary 

Let \(f(x)\) be a twice differentiable function on an open interval \(I\) and \(c\in I\).
  1. If \(f''(c) > 0\), then \(f(x)\) is convex at \(x=c\). 
  2. If \(f''(c) < 0\), then \(f(x)\) is concave at \(x=c\).
Proof. We prove only part 1. Part 2 is similar.

Let us define \(g(x) = f(x) - [f'(c)(x - c) + f(c)]\). \(g(x)\) is twice differentiable on \(I\) and \(g'(x) = f'(x) - f'(c)\) so \(g(c) = g'(c) = 0\). Also, \(g''(x) = f''(x)\) so that \(g''(c) = f''(c) > 0\). Therefore, by Corollary {cor:minmax}, \(x=c\) is a local minimum point of \(g(x)\). By the definition of local minimum, it follows that \(f(x)\) is convex \(x = c\). ■

Example. Consider \(f(x) = e^{-x^2}\). Let us find the range of \(x\) where \(f(x)\) is convex and where it is concave. Let us also find the extrema of \(f(x)\).
\[ \begin{eqnarray} f'(x) &=& -2xe^{-x^2},\\ f''(x) &=& -2e^{-x^2} + 4x^2e^{-x^2} = 2(2x^2 - 1)e^{-x^2}. \end{eqnarray} \]
Solving \(f''(x) = 0\) yields \(x = \pm\frac{1}{\sqrt{2}}\). Therefore,
  • \(f''(x) < 0\) if \(-\frac{1}{\sqrt{2}} < x < \frac{1}{\sqrt{2}}\) where \(f(x)\) is concave, and
  • \(f''(x) > 0\) if \(x < -\frac{1}{\sqrt{2}}\) or \(x > \frac{1}{\sqrt{2}}\) where \(f(x)\) is convex.
\(f'(x) = 0\) only if \(x = 0\) and \(f''(0) = -2 < 0\). So \(f(0)\) is a local maximum value (in fact, it is the global maximum value). □

As in the above example, the point at which the sign of the second-derivative changes is called an inflection point.

Comments

Post a Comment

Popular posts from this blog

Open sets and closed sets in \(\mathbb{R}^n\)

-
Image
Open sets In \(\mathbb{R}\), we have the notion of an open interval such as \((a, b) = \{x \in \mathbb{R} | a < x < b\}\). We want to extend this idea to apply to \(\mathbb{R}^n\). We also introduce the notions of bounded sets and closed sets in \(\mathbb{R}^n\). Recall that the \(\varepsilon\)-neighbor of a point \(x\in\mathbb{R}^n\) is defined as \(N_{\varepsilon}(x) = \{y \in \mathbb{R}^n | d(x, y) < \varepsilon \}\) where \(d(x,y)\) is the distance between \(x\) and \(y\). Definition (Open set) A subset \(U\) of \(\mathbb{R}^n\) is said to be an  open set  if the following holds: \[\forall x \in U ~ \exists \delta > 0 ~ (N_{\delta}(x) \subset U).\tag{Eq:OpenSet}\] That is, for every point in an open set \(U\), we can always find an open ball centered at that point, that is included in \(U\). See the following figure. Perhaps, it is instructive to see what is  not  an open set. Negating (Eq:OpenSet), we have \[\exists x \in U ~ \forall \delta > 0 ~ (N_{\delta}(x) \not
Read more

Euclidean spaces

-
Image
We would like to study multivariate functions (i.e., functions of many variables), continuous multivariate functions in particular. To define continuity, we need a measure of "closeness" between points. One measure of closeness is the Euclidean distance. The set \(\mathbb{R}^n\) (with \(n \in \mathbb{N}\)) with the Euclidean distance function is called a Euclidean space. This is the space where our functions of interest live. The real line is a geometric representation of \(\mathbb{R}\), the set of all real numbers. That is, each \(a \in \mathbb{R}\) is represented as the point \(a\) on the real line. The coordinate plane , or the \(x\)-\(y\) plane , is a geometric representation of \(\mathbb{R}^2\), the set of all pairs of real numbers. Each pair of real numbers \((a, b)\) is visualized as the point \((a, b)\) in the plane. Remark . Recall that \(\mathbb{R}^2 = \mathbb{R}\times\mathbb{R} = \{(x, y) | x, y \in \mathbb{R}\}\) is the Cartesian product of \(\mathbb{R}\) with i
Read more

Newton's method

-
Image
Newton's method (the Newton-Raphson method) is a very powerful numerical method for solving nonlinear equations.  Suppose we'd like to solve a non-linear equation \(f(x) = 0\), where \(f(x)\) is a (twice) differentiable non-linear function. Newton's method generates a sequence of numbers \(c_1, c_2, c_3, \cdots\) that converges to a solution of the equation. That is, if \(\alpha\) is a solution (i.e., \(f(\alpha) = 0\)) then,  \[\lim_{n\to\infty}c_n = \alpha,\] and this sequence \(\{c_n\}\) is generated by a series of linear approximations of the function \(f(x)\). Theorem (Newton's method) Let \(f(x)\) be a function that is twice differentiable on an open interval \(I\) that contains the closed interval \([a, b]\) (i.e., \([a,b]\subset I\)) and satisfy the following conditions: \(f(a) < 0\) and \(f(b) > 0\); For all \(x\in [a, b]\), \(f'(x) > 0\) and \(f''(x) > 0\). Let us define the sequence \(\{c_n\}\) by \[ \begin{eqnarray} c_1 &
Read more