The first derivative test

Once the stationary points have been found, the first derivative test establishes what type they are, by observing how the sign of $f'$ changes around each of them.

Property — Sufficient condition (first derivative test)

If $f'(x_0)=0$ and:

• $f'$ goes from $+$ to $-$ around $x_0$ $\implies$ relative maximum;
• $f'$ goes from $-$ to $+$ $\implies$ relative minimum;
• $f'$ does not change sign $\implies$ inflection with horizontal tangent (neither max nor min).

The idea is intuitive: where $f'>0$ the function increases, where $f'<0$ it decreases. If it first increases and then decreases we are at the top of a “hump” (maximum); if it first decreases and then increases we are at the bottom of a “valley” (minimum); if the sign does not change the function resumes the same monotonicity and the point is a horizontal inflection.

The three behaviours of a stationary point: maximum ($f'$ from $+$ to $-$), minimum ($f'$ from $-$ to $+$) and inflection with horizontal tangent ($f'$ does not change sign).

Links

Topics: Function study
Concepts: Increase and decrease · Inflection · Relative maxima and minima · Stationary point
Methods: Maximum minimum via derivative
Skills: Studying a function · Sketching a graph