Classify stationary points

Stationary does not automatically mean a turning point

A stationary point has $dy/dx=0$. Find its input first, then inspect the slope on both sides to classify it. Substitute the input into the original function when you need the point’s height.

In the source’s combined sketch, A is a maximum, B a minimum and C a stationary point of inflexion (also called inflection). Its separate sketches show positive-to-negative slope at A, negative-to-positive at B, and positive/positive or negative/negative slopes around a horizontal tangent at C.

A local maximum has the highest value within a sufficiently small neighbourhood; a local minimum has the lowest.

The Teacher specifically asks students to discuss “neighbourhood”: a local highest/lowest value need not be the highest/lowest value of the whole graph. The source’s strict-neighbourhood descriptions refer to strict local extrema.

A stationary inflection additionally changes concavity.

The source describes its gradient as zero with the same sign on either side, and its next sentence requires a concavity change. A same-sign first derivative alone rules out an extremum under these sign assumptions, but it does not, without a concavity check, establish every possible function’s inflection behaviour. Keep both requirements rather than treating the sign table as a converse theorem.

First derivative test near a stationary input a

Here $a^-$ and $a^+$ mean inputs just to the left and right, not powers. All source cases and tangent directions are retained:

Case 1

• f′ left: positive.
• f′ at a: zero.
• f′ right: negative.
• Tangent directions: rising → horizontal → falling.
• Source classification: Local maximum.

Case 2

• f′ left: negative.
• f′ at a: zero.
• f′ right: positive.
• Tangent directions: falling → horizontal → rising.
• Source classification: Local minimum.

Case 3, increasing

• f′ left: positive.
• f′ at a: zero.
• f′ right: positive.
• Tangent directions: rising → horizontal → rising.
• Source classification: Stationary inflection in the source’s sketch.

Case 3, decreasing

• f′ left: negative.
• f′ at a: zero.
• f′ right: negative.
• Tangent directions: falling → horizontal → falling.
• Source classification: Stationary inflection in the source’s sketch.

For the last two cases, confirm the concavity change as explained above. The mathematical reconstructions below show the source’s local configurations, including both stationary-inflection orientations.

Second derivative test at a stationary input

Assume the usual differentiability conditions near a and $f^{\prime}(a)=0$. The complete source test has three outcomes:

• Negative f″ at a: Local maximum.

• Positive f″ at a: Local minimum.

• Zero f″ at a: Inconclusive; use another test such as the first derivative test.

A zero second derivative is not proof of an inflection. The source recommends the first derivative test if the second derivative is tedious. Example 1 retains both analytical tests and the separate numerical calculator workflow.

Test a stationary point when the second derivative is zero

For $f(x)=x^3$ and $g(x)=x^4$, both first and second derivatives vanish at zero. Classify each stationary point using behaviour on both sides. Explain why a zero second derivative cannot supply the classification on its own.

After your attempt: compare the reasoning

For $f$, $f^{\prime}(x)=3x^2$ is positive on both sides of zero, so there is no local extremum. Also $f^{\prime\prime}(x)=6x$ changes sign, giving a stationary inflection. For $g$, $g^{\prime}(x)=4x^3$ changes from negative to positive, giving a minimum; $g^{\prime\prime}(x)=12x^2$ has no concavity sign change. The second derivative test is inconclusive when its value is zero.