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Using the First Derivative Test to Determine Relative (Local) Extrema AB & BC

At a critical point $c$ where $f$ is continuous, the First Derivative Test reads the sign of $f'$ from left to right: positive to negative gives a relative maximum, negative to positive gives a relative minimum, and no change gives neither. On $f(x) = 3x^{4} - 4x^{3} - 12x^{2} + 5$, whose derivative factors as $12x(x + 1)(x - 2)$, the four test values $-48$, $7.5$, $-24$, $144$ produce a minimum at $x = -1$, a maximum at $x = 0$, and a minimum at $x = 2$.

Two habits break the test. The direction gets reversed, and a rise followed by a fall is reported as a minimum. And a critical point is assumed to be an extremum without the sign ever being checked, which fails on $(x - 1)^{3} + 2$, where the tangent flattens at $x = 1$ and the function keeps climbing. The test needs no derivative at $c$ itself, so it settles the corner of $|x|$ and the cusp of $x^{2/3}$ as minima, and it does need $f$ to be continuous at $c$. Finally, which kind, where, and what value are three different answers, and the value comes from $f$ rather than $f'$.

AT A CRITICAL POINT, READ THE SIGN OF f′ LEFT TO RIGHT ACROSS IT. f′ > 0 f′ < 0 f′ < 0 f′ > 0 f′ > 0 f′ > 0 RELATIVE MAXIMUM RELATIVE MINIMUM NEITHER RISES THEN FALLS FALLS THEN RISES NO SIGN CHANGE f′ = 0 AND f′ UNDEFINED BOTH QUALIFY AS CRITICAL. WHAT DECIDES THE VERDICT IS THE SIGN CHANGE, AND NO CHANGE AT ALL IS ONE OF THE THREE OUTCOMES.
The third panel is a critical point that is not an extremum. Its tangent is as flat as the first two and its function never turns.
f(x) = 3x⁴ − 4x³ − 12x² + 5, f′(x) = 12x(x + 1)(x − 2) f(−1) = 0 f(0) = 5 f(2) = −27 f′ < 0 f′ > 0 f′ < 0 f′ > 0 x = −1 MIN x = 0 MAX x = 2 MIN THREE VERDICTS FROM THREE SIGN CHANGES, READ LEFT TO RIGHT.
Drawn to scale at 100 px per unit across and 3 px per unit up. The heights come from $f$; the verdicts come from the row of signs beneath it.

The work

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Lesson
Using the First Derivative Test to Determine Relative (Local) Extrema ›

Runs the First Derivative Test in the direction it is actually read, works a quartic with three critical points end to end, treats no sign change as a real verdict rather than a stall, shows the test settling corners and cusps where no derivative exists, and separates which kind of extremum from where it is and what its value is.

Skill check · 10 scenarios
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Diagnostic
10-item topic check ›

Ten items spanning the two failure modes of this topic: reading the sign change backwards so a peak is reported as a valley, and treating every critical point as an extremum without ever checking whether the derivative changed sign at all.

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Targeted Practice
Drill a single misconception ›

Pick one of the failure modes you missed and drill it on its own. The round is adaptive: two correct in a row clears the misconception and moves you to the next.

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