First Derivative Test

Key idea

Classify extrema by sign changes of f′.

In depth

At a critical point, if f′ changes from positive to negative, it is a local maximum; negative to positive, a local minimum; no sign change, neither. It reads the direction of travel before and after.

Worked example

f′(x) = 3x² − 3: it changes from + to − at −1 (max) and from − to + at 1 (min).

Real-world uses

• Locating the peak of a hill profile or the lowest point of a reservoir level

Common mistake

Testing the sign only on one side.

How to learn this

1. Read the key idea and say it in your own words.
2. Work through the worked example, then try a similar problem.
3. Use the interactive lesson’s visual and mini-quiz to check the pattern.

Frequently asked questions

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Related advanced topics in Calculus

• Idea of a Limit – The value a function approaches as the input approaches a point.
• One-Sided Limits – Left and right limits must agree for the limit to exist.
• Limit Laws – Limits of sums, products and quotients are computed from the parts.
• Factoring and Rationalizing Limits – Remove 0/0 indeterminacy with algebra.
• Limits at Infinity – End behaviour of functions.
• Infinite Limits and Asymptotes – Function values grow without bound near a vertical asymptote.