Increasing/decreasing test Calculator
Uses the sign of the derivative on an interval to decide whether a function is increasing or decreasing.
Formula first
Overview
Uses the sign of the derivative on an interval to decide whether a function is increasing or decreasing. It explains the calculus condition behind the rule and how that condition controls graph behavior. Students should use it to decide what can be concluded from derivatives, extrema, or interval hypotheses.
Symbols
Variables
result = result
Apply it well
When To Use
When to use: Use this when a calculus problem asks about monotonicity, concavity, local extrema, mean-value-theorem consequences, or indeterminate quotient forms.
Why it matters: These tests turn derivative information into clear statements about graph behavior and limits.
Avoid these traps
Common Mistakes
- Skipping a required continuity or differentiability condition.
- Using a one-point derivative value to conclude behavior on a whole interval.
One free problem
Practice Problem
If f'(x) > 0 for all x in the interval (1, 5), which of the following best describes the behavior of f(x) on that interval?
Solve for: result
Hint: Recall that a positive rate of change means the function values are rising.
The full worked solution stays in the interactive walkthrough.
References
Sources
- OpenStax, Calculus Volume 1, Section 4.5: Derivatives and the Shape of a Graph, accessed 2026-04-09
- Wikipedia: Monotonic function, accessed 2026-04-09
- Calculus, by James Stewart
- Introduction to Calculus and Analysis, by Richard Courant and Fritz John
- Thomas' Calculus