Increasing/Decreasing Functions and Local Extrema in Business Calculus

Graph Description and Function Behavior

Increasing and Decreasing Functions

Understanding where a function increases or decreases is fundamental in calculus, especially for analyzing business models and optimization problems.

• Increasing Function: A function f is increasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) < f(x2).

• Decreasing Function: A function f is decreasing on an interval if, for any two numbers x1 and x2 in the interval, x1 < x2 implies f(x1) > f(x2).

Theorem: For the interval (a, b):

• If f'(x) > 0 for all x in (a, b), then f is increasing on (a, b).

• If f'(x) < 0 for all x in (a, b), then f is decreasing on (a, b).

Graphical Interpretation: If the graph of f rises as you move from left to right, f is increasing; if it falls, f is decreasing.

Local Maximum and Minimum (Local Extrema)

A local maximum or local minimum (collectively called local extrema) occurs at a point where the function changes from increasing to decreasing (maximum) or from decreasing to increasing (minimum).

• Local Maximum: The highest point in a small neighborhood of the graph.

• Local Minimum: The lowest point in a small neighborhood of the graph.

Critical Points: Points where f'(x) = 0 or f'(x) does not exist are called critical values of f. Local extrema can only occur at critical points.

Theorem: Increasing and Decreasing Functions

Condition	f'(x)	Graph of f(x)	Examples
Increasing	> 0	Rises	↗
Decreasing	< 0	Falls	↘

Additional info: This table summarizes how the sign of the derivative relates to the function's behavior.

Analyzing Graphs for Extrema and Intervals

Identifying Intervals of Increase and Decrease

Given a graph of f(x), you can identify where the function is increasing or decreasing by observing the slope:

• Increasing: Where the graph moves upward as you go from left to right.

• Decreasing: Where the graph moves downward as you go from left to right.

Example Table: Intervals and Critical Points

Interval	Test Value	Sign of f'(x)	f(x) Behavior
(-∞, 1)	0	+	Increasing
(1, 3)	2	-	Decreasing
(3, ∞)	4	+	Increasing

Additional info: This table helps organize the analysis of function behavior based on the sign of the derivative in each interval.

Critical Points and Horizontal Tangents

Horizontal Tangent (f'(c) = 0)

At points where f'(c) = 0, the graph of f has a horizontal tangent. These points are candidates for local maxima or minima, but not all such points are extrema.

• If the sign of f'(x) changes from positive to negative at c, f(c) is a local maximum.

• If the sign of f'(x) changes from negative to positive at c, f(c) is a local minimum.

• If the sign does not change, there is no local extremum at c.

Critical Points Where f'(x) Does Not Exist

If f'(x) does not exist at a point but f(x) is defined, this point is still a critical point and may be a local extremum.

Procedure for Finding Local Extrema

1. Take the first derivative of the function.

2. Set f'(x) = 0 or find where f'(x) does not exist to determine critical numbers.

3. Make a chart of the derivative to organize intervals.

4. Test a point in each interval to determine the sign of f'(x).

5. Find local extrema by checking where the sign of f'(x) changes.

Example 1: Cubic Function

Given f(x) = x^3 - 6x^2 + 9x + 1:

• Find f'(x) = 3x^2 - 12x + 9.

• Solve f'(x) = 0 to find critical points: x = 1 and x = 3.

• Test intervals: (-∞,1), (1,3), (3,∞).

• Determine where f(x) is increasing or decreasing.

• Find local maximum at x = 1, local minimum at x = 3.

Example 2: Quartic Function

Given f(x) = x^4 + 2x^3 + 5:

• Find f'(x) = 4x^3 + 6x^2.

• Solve f'(x) = 0 to find critical points: x = 0 and x = -3/2.

• Test intervals: (-∞, -3/2), (-3/2, 0), (0, ∞).

• Determine where f(x) is increasing or decreasing.

• Find local minimum at x = -3/2.

Applications: Sketching and Interpreting Graphs

Sketching from Derivative Information

Given information about f(x) and f'(x) at specific points, you can sketch the graph of f(x) by:

• Plotting known points and slopes.

• Using the sign of f'(x) to determine intervals of increase and decrease.

• Identifying local maxima and minima where f'(x) changes sign.

Business Application Example: Price of Bacon

Given the rate of change of the price of bacon B(t) over time:

• Where B'(t) = 0, the price has a local extremum (maximum or minimum).

• Intervals where B'(t) > 0: price is increasing.

• Intervals where B'(t) < 0: price is decreasing.

By analyzing the sign of B'(t), you can describe the behavior of B(t) and sketch its graph, identifying local maxima and minima relevant to business decisions.

Summary Table: Relationship Between f'(x) and f(x)

f'(x)	Behavior of f(x)
> 0	Increasing
< 0	Decreasing
= 0 or DNE	Critical Point (possible extremum)