(L14) Intervals, Extrema, and the Mean Value Theorem in Business Calculus

Intervals and Their Types

Definition of Intervals

An interval is a connected subset of the real numbers ℝ. Intervals are fundamental in calculus for describing domains and regions of interest for functions.

• Open intervals: Do not include their endpoints.

• Closed intervals: Include their endpoints.

• Half-open (half-closed) intervals: Include only one endpoint.

• Unbounded intervals: Extend infinitely in one or both directions.

Examples of Intervals

• Open: $(-\infty, \infty)$, $(a, b)$

• Closed: $[a, b]$, $(-\infty, b]$, $[a, \infty)$

• Half-open: $[a, b)$, $(a, b]$

Intervals with infinite length are called unbounded, such as $(-\infty, a)$ or $(b, \infty)$.

Sign of Continuous Functions on Intervals: Test Points

Test Points and Sign Analysis

To determine where a continuous function is positive or negative, identify its zeros and use test points in each subinterval.

• Suppose $f$ is continuous and its only zeros are $a_1, a_2, ..., a_n$.

• These zeros partition the real line into subintervals.

• Pick a test point in each subinterval to determine the sign of $f$ there.

Example

Given $f(x) = x^5 - x$, factor to find zeros: $f(x) = x(x^4 - 1) = x(x-1)(x+1)(x^2+1)$.

• Zeros at $x = -1, 0, 1$ (since $x^2+1$ has no real roots).

• Test points: $-2, -1/2, 1/2, 2$.

• Evaluate $f$ at each test point to determine sign on each interval.

Table: Sign Analysis for $f(x) = x^5 - x$

Interval	x+1	x	x-1	x^2+1	f(x)
$(-\infty, -1)$	-	-	-	+	-
$(-1, 0)$	+	-	-	+	+
$(0, 1)$	+	+	-	+	-
$(1, \infty)$	+	+	+	+	+

Absolute and Relative Extrema

Definitions

• Absolute (Global) Maximum: $f(c)$ is the absolute maximum of $f$ on $S$ if $f(c) \geq f(x)$ for all $x \in S$.

• Absolute (Global) Minimum: $f(c)$ is the absolute minimum of $f$ on $S$ if $f(c) \leq f(x)$ for all $x \in S$.

• Relative (Local) Maximum: $f(c)$ is a local maximum if there is an open interval $(a, b)$ containing $c$ such that $f(c) \geq f(x)$ for all $x$ in $(a, b)$.

• Relative (Local) Minimum: $f(c)$ is a local minimum if there is an open interval $(a, b)$ containing $c$ such that $f(c) \leq f(x)$ for all $x$ in $(a, b)$.

Absolute and relative extrema are also called extreme values of $f$.

Key Properties

• Absolute/relative extrema may or may not exist for a function.

• On a given set, there can be at most one absolute maximum and one absolute minimum.

• Multiple relative extrema can exist.

Example (Graphical)

Given a function $f$ graphed on $[0, d]$, identify absolute and relative extrema by examining the highest and lowest points (absolute) and local peaks and valleys (relative).

Fermat's Theorem and Critical Numbers

Fermat's Theorem

Fermat's Theorem: If $f$ has a local maximum or minimum at $c$, and if $f'(c)$ exists, then $f'(c) = 0$.

This means that local extrema occur at points where the derivative is zero (horizontal tangent) or does not exist.

Critical Numbers

• A critical number of $f$ is a number $c$ in the domain of $f$ such that $f'(c) = 0$ or $f'(c)$ does not exist.

• Critical numbers are candidates for local extrema.

Cautions

• If $f'(c) = 0$, $f$ may not have a local extremum at $c$ (e.g., $f(x) = x^3$ at $x=0$).

• Local extrema can occur where $f'(c)$ does not exist (e.g., $f(x) = |x|$ at $x=0$).

Extreme Value Theorem

Extreme Value Theorem: If $f$ is continuous on a closed and bounded interval $[a, b]$, then $f$ attains an absolute maximum and minimum value on $[a, b]$.

Rolle's Theorem and the Mean Value Theorem (MVT)

Rolle's Theorem

Rolle's Theorem: If $f$ is continuous on $[a, b]$, differentiable on $(a, b)$, and $f(a) = f(b)$, then there is at least one $c$ in $(a, b)$ such that $f'(c) = 0$.

Illustration

• Rolle's Theorem guarantees a horizontal tangent (slope zero) somewhere between $a$ and $b$ if the function starts and ends at the same value.

Mean Value Theorem (MVT)

Mean Value Theorem: If $f$ is continuous on $[a, b]$ and differentiable on $(a, b)$, then there exists $c$ in $(a, b)$ such that

$f'(c) = \frac{f(b) - f(a)}{b - a}$

or equivalently,

$f(b) - f(a) = f'(c)(b - a)$

This means that at some point, the instantaneous rate of change (the derivative) equals the average rate of change over the interval.

Applications

• Used to prove properties about increasing/decreasing behavior of functions.

• Provides a link between the derivative and the overall change in a function.

Summary Table: Types of Extrema

Type	Definition	Where it Occurs
Absolute Maximum	$f(c) \geq f(x)$ for all $x$ in $S$	Interior or endpoint
Absolute Minimum	$f(c) \leq f(x)$ for all $x$ in $S$	Interior or endpoint
Relative Maximum	$f(c) \geq f(x)$ for all $x$ near $c$	Interior
Relative Minimum	$f(c) \leq f(x)$ for all $x$ near $c$	Interior

Key Formulas

• Average Rate of Change: $\frac{f(b) - f(a)}{b - a}$

• Critical Number: $f'(c) = 0$ or $f'(c)$ does not exist

• Mean Value Theorem: $f'(c) = \frac{f(b) - f(a)}{b - a}$

Example: Applying the Mean Value Theorem

Suppose $f(1) = 2$ and $f(5) = 10$. By the Mean Value Theorem, there exists $c$ in $(1, 5)$ such that

$f'(c) = \frac{10 - 2}{5 - 1} = 2$

This means that at some point between $x=1$ and $x=5$, the instantaneous rate of change of $f$ is exactly 2.