Understanding Increasing and Decreasing Functions

In differential calculus, understanding how a function's output changes with respect to its input is crucial. This involves identifying whether a function is increasing or decreasing over a given interval. The first derivative of a function provides a powerful tool to determine this behavior.

Defining Increasing and Decreasing Functions

A function $f(x)$ is said to be increasing on an interval if for any two numbers $x_1$ and $x_2$ in the interval, where $x_1 < x_2$ , we have $f(x_1) < f(x_2)$ . Conversely, a function $f(x)$ is decreasing on an interval if for any two numbers $x_1$ and $x_2$ in the interval, where $x_1 < x_2$ , we have $f(x_1) > f(x_2)$ .

The sign of the first derivative tells us if a function is increasing or decreasing.

If the first derivative $f'(x)$ is positive on an interval, the function $f(x)$ is increasing on that interval. If $f'(x)$ is negative, the function is decreasing.

The relationship between a function and its derivative is fundamental. For a differentiable function $f(x)$ , if $f'(x) > 0$ for all $x$ in an open interval $(a, b)$ , then $f(x)$ is strictly increasing on $(a, b)$ . If $f'(x) < 0$ for all $x$ in $(a, b)$ , then $f(x)$ is strictly decreasing on $(a, b)$ . If $f'(x) = 0$ at a point, it might indicate a local maximum, minimum, or an inflection point where the function momentarily stops increasing or decreasing.

What does a positive first derivative (

f'(x) > 0

) indicate about a function

f(x)

on an interval?

It indicates that the function $f(x)$ is increasing on that interval.

Finding Intervals of Increase and Decrease

To find the intervals where a function is increasing or decreasing, we follow a systematic process:

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Critical points are points where the derivative is either zero or undefined. These points divide the number line into intervals. We then pick a test value within each interval and substitute it into the first derivative, $f'(x)$ , to determine its sign. If $f'(x)$ is positive, the function is increasing in that interval. If $f'(x)$ is negative, the function is decreasing.

Remember that critical points themselves are not part of the open intervals of increase or decrease, but they mark the boundaries where the behavior of the function can change.

Example: Analyzing $f(x) = x^3 - 6x^2 + 5$

Let's apply the steps to the function $f(x) = x^3 - 6x^2 + 5$ .

Find the derivative: $f'(x) = 3x^2 - 12x$ .
Find critical points: Set $f'(x) = 0$ . $3x^2 - 12x = 0 \implies 3x(x - 4) = 0$ . Critical points are $x=0$ and $x=4$ .
Create intervals: The critical points divide the number line into three intervals: $(-\infty, 0)$ , $(0, 4)$ , and $(4, \infty)$ .
Test values:
- For $(-\infty, 0)$ , let's test $x=-1$ : $f'(-1) = 3(-1)^2 - 12(-1) = 3 + 12 = 15 > 0$ .
- For $(0, 4)$ , let's test $x=2$ : $f'(2) = 3(2)^2 - 12(2) = 12 - 24 = -12 < 0$ .
- For $(4, \infty)$ , let's test $x=5$ : $f'(5) = 3(5)^2 - 12(5) = 75 - 60 = 15 > 0$ .
Determine intervals of increase/decrease:
- $f(x)$ is increasing on $(-\infty, 0)$ and $(4, \infty)$ because $f'(x) > 0$ .
- $f(x)$ is decreasing on $(0, 4)$ because $f'(x) < 0$ .

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Monotonicity and Extrema

The intervals of increase and decrease are directly related to the local extrema (local maximum and minimum) of a function. A function typically changes from increasing to decreasing at a local maximum, and from decreasing to increasing at a local minimum. These changes occur at critical points where the derivative is zero or undefined.

Sign of f'(x)	Behavior of f(x)
$f'(x) > 0$	Increasing
$f'(x) < 0$	Decreasing
$f'(x) = 0$ or undefined	Potential local extremum or inflection point

Learning Resources

Khan Academy: Increasing and Decreasing Functions(video)

A clear video explanation of how the first derivative determines if a function is increasing or decreasing, with examples.

Paul's Online Math Notes: Increasing/Decreasing Functions(documentation)

Comprehensive notes covering the definition, theorems, and step-by-step examples for finding intervals of increase and decrease.

Calculus: Increasing and Decreasing Functions - YouTube(video)

A visual tutorial demonstrating how to find intervals of increase and decrease for various functions.

Math is Fun: Increasing and Decreasing Functions(blog)

An accessible explanation of the concepts with interactive graphs and simple examples.

Brilliant.org: Monotonicity(documentation)

Explains monotonicity and its relation to increasing and decreasing functions, often with interactive elements.

University of British Columbia: Increasing and Decreasing Functions(paper)

Lecture notes providing a formal definition and theorems related to increasing and decreasing functions.

Symbolab: Increasing and Decreasing Functions Calculator(documentation)

An online tool to help verify your answers by calculating intervals of increase and decrease for a given function.

Coursera: Calculus Specialization - Increasing/Decreasing Functions(video)

Part of a broader calculus course, this lecture focuses specifically on identifying function behavior using derivatives.

Wikipedia: Monotonic Function(wikipedia)

Provides a formal mathematical definition and properties of monotonic functions, including increasing and decreasing cases.

Art of Problem Solving: Increasing and Decreasing Functions(documentation)

A resource often used for competitive math preparation, offering clear explanations and problem-solving strategies.