How to Determine Where a Function Is Increasing or Decreasing

A differentiable function is increasing on intervals where its first derivative is positive and decreasing where its first derivative is negative. The key is to analyze intervals, not isolated test points.

Derivative sign rule

• If f′(x)>0 on an interval, f is increasing there.
• If f′(x)<0 on an interval, f is decreasing there.
• If f′(x)=0 at an isolated point, the behavior depends on the sign on both sides.

Worked example

Let f(x)=x^3-3x. Then f′(x)=3(x-1)(x+1). The derivative is zero at x=-1 and x=1, which split the real line into three intervals.

Choose one test value in each interval:

• At x=-2, f′(-2)>0.
• At x=0, f′(0)<0.
• At x=2, f′(2)>0.

Therefore, the function increases on (-∞,-1) and (1,∞), and decreases on (-1,1).

Why critical numbers are not intervals

Do not write that the function is increasing “at x=-1.” Increase and decrease describe behavior over an interval. Critical points are boundaries between intervals or special points that require further analysis.

Domain breaks must be included

If the function is undefined at a value, split the sign analysis there even if the derivative equation does not highlight it. For 1/(x-1), the domain break at x=1 separates two decreasing branches.

What if the derivative does not exist?

A function can have a local extremum at a corner or cusp where the derivative is undefined. The absolute-value function |x| decreases for x<0 and increases for x>0, with a local minimum at x=0.

Use a sign chart

A first-derivative sign chart places the critical numbers in order and records + or - on each interval. The Variation Table tool can produce either a full variation table or a sign-focused table.

Common mistakes

• Solving f′(x)=0 but never checking signs.
• Testing a point outside the function domain.
• Including critical endpoints inside open intervals.
• Assuming every stationary point is a maximum or minimum.
• Ignoring points where the derivative is undefined.

Analyze increasing and decreasing intervals online →

For a table centered on derivative signs, continue with creating a first-derivative sign chart.