AD - Extrema and the Extreme Value Theorem Lesson

Extrema and the Extreme Value Theorem

Developing an intuitive understanding of local and global extrema is an important first step in learning how the behavior of a function is shaped by extrema. View the presentation relating maxima/minima and slope with a function's graph.

Absolute (Global) Maxima and Minima

A function f has an absolute maximum (or global maximum) at c if f(c) > f(x) for all x in D, where D is the domain of f. The number f(c) is called the maximum value of f on D. Similarly, f has an absolute minimum (or global minimum) at c if f(c) < f(x) for all x in D and the number f(c) is called the minimum value of f on D. The maximum and minimum values of f are referred to as the extreme values of f. View the presentations below on absolute extrema.

Extreme Value Theorem

Conditions under which a function is guaranteed to possess extreme values are given by the Extreme Value Theorem: If f is continuous on a closed interval [a, b], then f attains an absolute minimum value f(c) and an absolute maximum value f(d) at some numbers c and d in the closed interval [a, b]such that f(c) < f(x) < f(d) for all x in [a, b].

Although some functions have extreme values, others do not as illustrated below. It is possible that a particular extreme value can be attained more than once in the closed interval.

Functions with Extreme Values	Functions without Extreme Values
$LaTeX: f\left(x\right)=\frac{\left(\sin x\right)}{x}$ $f\left(x\right)=\frac{\left(\sin x\right)}{x}$ Absolute maximum at (0, 1) Absolute minima at (-4.493, -0.217) and (4.493, -0.217) Absolute maximum value: 1 Absolute minimum value: -0.217	$LaTeX: f\left(x\right)=\frac{1}{x^4}$ $f\left(x\right)=\frac{1}{x^4}$ (Not continuous nor differentiable at x = 0)
$LaTeX: f\left(x\right)=-0.6\left(x+3\right)\left(x\right)\left(x-1\right)\left(x-4\right)$ $f\left(x\right)=-0.6\left(x+3\right)\left(x\right)\left(x-1\right)\left(x-4\right)$ Absolute maxima at (-2, 21,6) and (3, 21.6) Absolute maximum value: 21.6	$LaTeX: f\left(x\right)=\left(x-3\right)^{\frac{1}{3}}$ $f\left(x\right)=\left(x-3\right)^{\frac{1}{3}}$ (Continuous but not differentiable at x = 3)

Functions with Extreme Values

Functions without Extreme Values

equation image indicator $LaTeX: f\left(x\right)=\frac{\left(\sin x\right)}{x}$ $f\left(x\right)=\frac{\left(\sin x\right)}{x}$

graph of sinx / x

Absolute maximum at (0, 1)

Absolute minima at (-4.493, -0.217)

and (4.493, -0.217)

Absolute maximum value: 1

Absolute minimum value: -0.217

$LaTeX: f\left(x\right)=\frac{1}{x^4}$ $f\left(x\right)=\frac{1}{x^4}$ equation image indicator

graph of 1 / x ^ (1/4)

(Not continuous nor differentiable at x = 0)

equation image indicator $LaTeX: f\left(x\right)=-0.6\left(x+3\right)\left(x\right)\left(x-1\right)\left(x-4\right)$ $f\left(x\right)=-0.6\left(x+3\right)\left(x\right)\left(x-1\right)\left(x-4\right)$

graph of Rolles Theorem

Absolute maxima at (-2, 21,6) and (3, 21.6)

Absolute maximum value: 21.6

equation image indicator $LaTeX: f\left(x\right)=\left(x-3\right)^{\frac{1}{3}}$ $f\left(x\right)=\left(x-3\right)^{\frac{1}{3}}$

graph of continuous not differ

(Continuous but not differentiable at x = 3)

Relative (Local) Maxima and Minima

A function f has a relative maximum (or local maximum) at an interior point c within its domain D if f(c) > f(x) for all x in some open interval containing c. A function f has a relative minimum (or local minimum) at an interior point c within its domain D if f(c) < f(x) for all x in some open interval containing c.

View two presentations related to finding relative extrema.

Critical Number

A critical number of a function f is a number c in the domain of f such that either f'(c) = 0 or f'(c) does not exist. The only points where a function can take on extreme values are critical points and endpoints. Critical numbers partition the x-axis into intervals on which f' is either positive or negative. It is helpful to record the behavior of f between and at critical points using a sign chart.

Extrema and the Extreme Value Theorem Practice

Extrema and the Extreme Value Theorem: Even More Problems!

Complete problems from your textbook and/or online resources as needed to ensure your complete understanding of extrema and the Extreme Value Theorem.

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