Local maximum and local minimum

-

Derivatives can be used to find a function's local extremal values (i.e., local maximum or minimum values).

If \(f(x)\) is a continuous function defined on a closed interval \([a,b]\), then it always has a maximum value and a minimum value (c.f. the Extreme Value Theorem). Finding a maximum or minimum of a function on its entire domain is a global problem.

On the other hand, examining continuity or differentiability around \(x=a\) is a local problem in the sense that it only concerns the neighbor of the point \(x = a\). We may also define the notions of local maxima and local minima.



Definition (Local maximum and local minimum)

Let \(f(x)\) be a function defined on an interval \(I\) and let \(a \in I\).

  1. \(f(a)\) is said to be a local maximum value of the function \(f(x)\) if there exists  \(\delta > 0\) such that, for all \(x\in (a - \delta, a + \delta) \cap I\), \(x \neq a\) implies \(f(x) < f(a)\). \(x= a\) is said to be a local maximum point if \(f(a)\) is a local maximum value.
  2. \(f(a)\) is said to be a local minimum value of the function \(f(x)\) if there exists  \(\delta > 0\) such that, for all \(x\in (a - \delta, a + \delta) \cap I\), \(x \neq a\) implies \(f(x) > f(a)\). \(x= a\) is said to be a local minimum point if \(f(a)\) is a local minimum value.
We use the term extreme value to mean either a local maximum value or a local minimum value.

Remark. If a local maximum value is actually a maximum value, then the maximum value is called a global maximum value. A global minimum value is similarly defined. Note that a global maximum (minimum) value is a local maximum (minimum) value, but not vice versa. □

Example. The constant function \(f(x) = c\) (\(c\) is a constant) has the maximum value \(c\) and the minimum value \(c\), but it does not have a local maximum value or local minimum value. □

Example. Let \(f(x) = (x^2 - 1)(x^2 - 4)\) (draw the graph!). \(f(0) = 4\) is a local maximum value. In fact, take \(\delta = \sqrt{5} > 0\), then for all \(x\) such that \(0 < |x| < \delta\), we have
\[0  < x^2 < \delta^2\]
(i.e., \(x^2\) is strictly positive) and
\[-5  < x^2 - 5 < \delta^2 - 5 = 0\]
(i.e., \(x^2 - 5\) is strictly negative) so that
\[f(x) = x^2(x^2 - 5) + 4 < 4 = f(0).\]
This means that \(f(0) > f(x)\) for all \(x \in (-\delta, +\delta), x \neq 0\). That is, \(f(0)\) is a local maximum value (check the above definition again!). □

Definition (Stationary/critical point)

Let \(f(x)\) be a differentiable function. \(x=a\) is said to be a critical point or stationary point if \(f'(a) = 0\). In this case, \(f(a)\) is called a critical value or stationary value.

Theorem

If a differentiable function \(f(x)\) has a local extremum value (i.e., either a local maximum or a local minimum value) at \(x = a\), then \(x = a\) is a stationary point.
Proof. We prove only for the local maximum case (the local minimum case is similar).
Suppose \(f(x)\) takes a local maximum value at \(x=a\). Then there exists \(\delta > 0\) such that, for all \(x\in (a - \delta, a + \delta)\), \(f(x) \leq f(a)\). Since \(f(x)\) is differentiable at \(x = a\) (by assumption), we have
\[ \begin{eqnarray} f'(a) &=& \lim_{x\to a}\frac{f(x) - f(a)}{x - a}\\ &=&\lim_{x\to a+0}\frac{f(x) - f(a)}{x - a}\\ &=&\lim_{x\to a-0}\frac{f(x) - f(a)}{x - a}. \end{eqnarray} \]
  • If \(0 < a - x < \delta\) (i.e., \(x \in (a - \delta, a)\)), then \(\frac{f(x) - f(a)}{x - a} \geq 0\) so, by taking the limit as \(x \to a-0\), we have \(f'(a)\geq 0\). 
  • If \(0 < x - a < \delta\) (i.e., \(x\in (a, a+\delta)\)), then \(\frac{f(x) - f(a)}{x - a} \leq 0\) so, by taking the limit as \(x \to a+0\), we have \(f'(a) \leq 0\). 
Therefore, we have \(f'(a) = 0\). ■
Remark. The converse of this theorem is not true. That is, \(f'(a) = 0\) does not imply \(f(a)\) is an extremum. For example, \(f(x) = x^3\) has \(f'(0) = 0\), but \(f(0)\) is neither a local maximum value nor a local minimum value (draw the graph!). □

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