MathematicsLocal extreme valuesUniversity
IBUndergraduate

First derivative test

Classifies local extrema by how the sign of the first derivative changes around a critical number.

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Core idea

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.

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.

Symbols

Variables

result = result

result
result
Variable

Walkthrough

Derivation

Derivation of First derivative test

Classifies local extrema by how the sign of the first derivative changes around a critical number.

  • f is continuous at the critical number c.
  • The sign of f' is known on both sides of c.
1

State the verified result

This is the standard calculus statement used for this entry.

2

Use the hypotheses

The conclusion is valid only under the stated assumptions.

Result

Source: OpenStax, Calculus Volume 1, Section 4.5: Derivatives and the Shape of a Graph, accessed 2026-04-09

Free formulas

Rearrangements

Solve for

Use First derivative test

Check the hypotheses, then apply the definition or derivative test exactly as stated.

Difficulty: 2/5

The static page shows the finished rearrangements. The app keeps the full worked algebra walkthrough.

Visual intuition

Graph

Graph unavailable for this formula.

Reaction-style formula

Why it behaves this way

Intuition

Read the graph from left to right: derivative signs describe rising or falling, second derivative signs describe bending, and indeterminate forms warn that a limit cannot be read from substitution alone.

function
The rule whose values or derivatives are being studied.
interval
The set over which behavior is being claimed.
f'
first derivative
Slope or instantaneous rate of change.

Signs and relationships

  • =>: The hypotheses imply the conclusion; the implication is not automatically reversible.

Free study cues

Insight

Canonical usage

The first derivative test is used to determine if a critical number corresponds to a local maximum or minimum by observing the change in the sign of the derivative.

Common confusion

Confusing the sign change of the derivative with the sign of the function's value itself.

Dimension note

The first derivative test itself does not have units. It analyzes the sign changes of the derivative of a function, which is a rate of change.

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?

sign_change+ to -

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.

Where it shows up

Real-World Context

Motion, cost, and growth models use derivative signs to identify whether quantities are rising, falling, leveling, or bending.

Study smarter

Tips

  • Check the interval first.
  • Check all hypotheses before using the conclusion.
  • Do not treat a theorem statement as a numeric calculator.

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.

Common questions

Frequently Asked Questions

Classifies local extrema by how the sign of the first derivative changes around a critical number.

Use this when a calculus problem asks about monotonicity, concavity, local extrema, mean-value-theorem consequences, or indeterminate quotient forms.

These tests turn derivative information into clear statements about graph behavior and limits.

Skipping a required continuity or differentiability condition. Using a one-point derivative value to conclude behavior on a whole interval.

Motion, cost, and growth models use derivative signs to identify whether quantities are rising, falling, leveling, or bending.

Check the interval first. Check all hypotheses before using the conclusion. Do not treat a theorem statement as a numeric calculator.

References

Sources

  1. OpenStax, Calculus Volume 1, Section 4.5: Derivatives and the Shape of a Graph, accessed 2026-04-09
  2. Wikipedia: Derivative test, accessed 2026-04-09
  3. Calculus by Michael Spivak
  4. Stewart's Calculus
  5. Introduction to Real Analysis by Robert G. Bartle and Donald R. Sherbert