Skill: Determine intervals where a function is increasing or decreasing

A function is:

• Increasing on an interval if its $y$-values rise as $x$ increases.
• Decreasing on an interval if its $y$-values fall as $x$ increases.

When a formula is given, use the first derivative:

• If $f'(x)>0$, then $f$ is increasing.
• If $f'(x)<0$, then $f$ is decreasing.

Worked example

Determine where

$$f(x)=x^3-3x^2+2$$

is increasing and decreasing.

1. Find the derivative

Differentiate the function:

$$f'(x)=3x^2-6x$$

Factor:

$$f'(x)=3x(x-2)$$

2. Find the critical numbers

Critical numbers occur where $f'(x)=0$ or where $f'(x)$ is undefined.

Solve:

$$3x(x-2)=0$$

So,

$$x=0 \quad \text{or} \quad x=2$$

The derivative is defined for every real number, so these are the only critical numbers.

They divide the number line into three intervals:

$$(-\infty,0), \qquad (0,2), \qquad (2,\infty)$$

3. Determine the sign of $f'(x)$

Choose one test value from each interval.

Interval	Test value	Sign of $f'(x)=3x(x-2)$	Behavior
$(-\infty,0)$	$x=-1$	positive	Increasing
$(0,2)$	$x=1$	negative	Decreasing
$(2,\infty)$	$x=3$	positive	Increasing

For example, at $x=1$:

$$f'(1)=3(1)(1-2)=-3<0$$

so the function is decreasing on $(0,2)$.

4. State the answer

The function is increasing on

$$(-\infty,0)\cup(2,\infty)$$

and decreasing on

$$(0,2)$$

The points $x=0$ and $x=2$ separate the intervals. They are not usually included in the interval notation for increasing or decreasing behavior because the behavior changes at those points. A derivative of zero identifies a possible turning point, but the sign chart is what confirms whether the function changes from increasing to decreasing or vice versa.