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# 数学代写|线性规划代写Linear Programming代考|Extreme Values of Functions on Closed Intervals

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## 数学代写|线性规划代写Linear Programming代考|Extreme Values of Functions on Closed Intervals

This section shows how to locate and identify extreme (maximum or minimum) values of a function from its derivative. Once we can do this, we can solve a variety of optimization problems (see Section 4.5). The domains of the functions we consider are intervals or unions of separate intervals.
DEFINITIONS Let $f$ be a function with domain $D$. Then $f$ has an absolute maximum value on $D$ at a point $c$ if
$$f(x) \leq f(c) \quad \text { for all } x \text { in } D$$
and an absolute minimum value on $D$ at $c$ if
$$f(x) \geq f(c) \quad \text { for all } x \text { in } D \text {. }$$
Maximum and minimum values are called extreme values of the function $f$. Absolute maxima or minima are also referred to as global maxima or minima.

For example, on the closed interval $[-\pi / 2, \pi / 2]$ the function $f(x)=\cos x$ takes on an absolute maximum value of 1 (once) and an absolute minimum value of 0 (twice). On the same interval, the function $g(x)=\sin x$ takes on a maximum value of 1 and a minimum value of -1 (Figure 4.1).

Functions defined by the same equation or formula can have different extrema (maximum or minimum values), depending on the domain. A function might not have a maximum or minimum if the domain is unbounded or fails to contain an endpoint. We see this in the following example.

## 数学代写|线性规划代写Linear Programming代考|Local (Relative) Extreme Values

Figure 4.5 shows a graph with five points where a function has extreme values on its domain $[a, b]$. The function’s absolute minimum occurs at $a$ even though at $e$ the function’s value is smaller than at any other point nearby. The curve rises to the left and falls to the right around $c$, making $f(c)$ a maximum locally. The function attains its absolute maximum at $d$. We now define what we mean by local extrema.
DEFINITIONS A function $f$ has a local maximum value at a point $c$ within its domain $D$ if $f(x) \leq f(c)$ for all $x \in D$ lying in some open interval containing $c$.
A function $f$ has a local minimum value at a point $c$ within its domain $D$ if $f(x) \geq f(c)$ for all $x \in D$ lying in some open interval containing $c$.
If the domain of $f$ is the closed interval $[a, b]$, then $f$ has a local maximum at the endpoint $x=a$ if $f(x) \leq f(a)$ for all $x$ in some half-open interval $[a, a+\delta), \delta>0$. Likewise, $f$ has a local maximum at an interior point $x=c$ if $f(x) \leq f(c)$ for all $x$ in some open interval $(c-\delta, c+\delta), \delta>0$, and a local maximum at the endpoint $x=b$ if $f(x) \leq f(b)$ for all $x$ in some half-open interval $(b-\delta, b], \delta>0$. The inequalities are reversed for local minimum values. In Figure 4.5, the function $f$ has local maxima at $c$ and $d$ and local minima at $a, e$, and $b$. Local extrema are also called relative extrema. Some functions can have infinitely many local extrema, even over a finite interval. One example is the function $f(x)=\sin (1 / x)$ on the interval $(0,1]$. (We graphed this function in Figure 2.40.)

An absolute maximum is also a local maximum. Being the largest value overall, it is also the largest value in its immediate neighborhood. Hence, a list of all local maxima will automatically include the absolute maximum if there is one. Similarly, a list of all local minima will include the absolute minimum if there is one.

## 数学代写|线性规划代写Linear Programming代考|Extreme Values of Functions on Closed Intervals

$$f(x) \leq f(c) \quad \text { for all } x \text { in } D$$

$$f(x) \geq f(c) \quad \text { for all } x \text { in } D \text {. }$$

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