First derivative test Calculator
Classifies local extrema by how the sign of the first derivative changes around a critical number.
Formula first
Overview
Classifies local extrema by how the sign of the first derivative changes around a critical number. 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) is positive for x < c and negative for x > c, what occurs at the critical point c?
Solve for: result
Hint: Consider whether the function is increasing or decreasing on either side of c.
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: Derivative test, accessed 2026-04-09
- Calculus by Michael Spivak
- Stewart's Calculus
- Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert