f vs. f'

When we discuss where a function is increasing or decreasing we mean for which $x$ -values is a function increasing or decreasing.

If we have a function of time, we might discuss when a function is increasing or decreasing, and we are talking about for which $t$ -values is a function increasing or decreasing.

Increasing/Decreasing Test

If $f '(x) > 0$ on an open interval, then $f$ is increasing on the interval.
If $f '(x) < 0$ on an open interval, then $f$ is decreasing on the interval.

DO: Ponder the graphs in the box above until you are confident of why the two conditions listed are true.

You should completely master this concept; a helpful shorthand:
$f'>0\Longleftrightarrow f \uparrow$ $f'<0 \Longleftrightarrow f\downarrow$

We use this test in several ways. In order to determine whether a function is increasing at a point $x=a$ , you only need to see if $f'(a)$ is positive. If you wish to know all places where a function increases and decreases, you must find the sign of the derivative for any values of $x$ .

DO: What do we know about whether $f$ is increasing or decreasing at $x=a$ if $f'(a)=0$ ?

Finding the intervals where $f'(x)$ is positive (or negative), and hence where $f(x)$ is increasing (or decreasing) is closely related to critical numbers.

Critical Numbers

Recall that a critical number (also called a critical point) is a value of $x$ where $f$ is defined, and where $f'(x)$ is either zero or doesn't exist. We have already seen that critical numbers are the only $x$ -values that can be local maxima or minima. They have a second property that is almost as important:

$f'$ can only change sign at a critical number.

The reason is simple. If

$f'(x)$ is continuous and it changes sign, then it has to pass through 0 on its way from negative to positive (or vice versa). That's the Intermediate Value Theorem. If

$f'(x)$ is not continuous where it changes sign, then that is a point where

$f'(x)$ doesn't exist. Either way,

$f'$ changes sign only at a critical number.

Thus, $f$ can change direction only at a critical number.

We will use critical numbers to find the intervals where $f(x)$ is increasing and decreasing.

The Six Pillars of Calculus

Trigonometry Review

Exponential Functions

Logarithms and Inverse functions

Intro to Limits

Limit Laws and Computations

Continuity and the Intermediate Value Theorem

Limits at Infinity

Rates of Change

The Derivative Function

Basic Differentiation Rules

Product and Quotient Rules

Derivatives of Trig Functions

The Chain Rule

Implicit Differentiation

Derivatives of Logs

Derivatives in Science

Related Rates

Linear Approximation and Differentials

Differentiation Review

Absolute and Local Extrema

The Mean Value and other Theorems

ff vs. f′f'

Indeterminate Forms and L'Hospital's Rule

Optimization

Newton's Method

Anti-derivatives

The Area under a curve

Definite Integrals

The Fundamental Theorem of Calculus

Increasing/Decreasing Test and Critical Numbers

How can we tell if a function is increasing or decreasing?

Critical Numbers

$f$ vs. $f'$