Second Derivative Test for Local Extrema: A Clear Guide

The second derivative test for local extrema classifies a critical point of a function as a local maximum, a local minimum, or neither, based on the sign of the second derivative at that point. If $f’© = 0$ and $f’‘© > 0$, the function has a local minimum at $c$; if $f’‘© < 0$, it has a local maximum; and if $f’'© = 0$, the test is inconclusive. That last case matters as much as the first two, and we’ll return to it.

The logic is geometric. The first derivative tells you where the tangent line is flat; the second derivative tells you how the curve is bending at that spot. A positive second derivative means the curve is concave up (like a cup), so a flat point there must sit at the bottom of a dip. A negative second derivative means concave down (like a hill), so a flat point sits at a peak. This is the same reasoning covered in standard calculus courses and reference notes such as Paul’s Online Math Notes.

How the Test Works

A critical point is any $x$-value in the domain where $f’(x) = 0$ or where $f’(x)$ does not exist. The second derivative test applies only to points where $f’(x) = 0$ — flat points, not corners or cusps. So the first job is always to differentiate, set the derivative equal to zero, and solve. Then, for each critical point $c$, compute $f’'©$.

The third row is where students most often slip. A value of zero does not mean “no extremum” and it does not mean “inflection point.” It means this particular test has nothing to say, and you need a different tool. When $f’‘© = 0$, fall back to the first derivative test: examine the sign of $f’(x)$ just to the left and just to the right of $c$. If $f’$ switches from negative to positive, there’s a local minimum. If it switches from positive to negative, there’s a local maximum. If it doesn’t change sign, there is no extremum — the flat point is a horizontal inflection point, as with $f(x) = x^3$ at $x = 0$, where $f’(0) = 0$ and $f’'(0) = 0$ but the function simply passes through flat.

A Worked Example

Consider $f(x) = x^3 - 3x$.

First derivative: $f’(x) = 3x^2 - 3$. Setting this to zero gives $3x^2 = 3$, so $x^2 = 1$, and the critical points are $x = -1$ and $x = 1$.

Second derivative: $f’'(x) = 6x$, a single, simple expression that makes evaluation easy.

Evaluate:

• At $x = -1$: $f’'(-1) = -6 < 0$, so there is a local maximum. The function value is $f(-1) = -1 + 3 = 2$.
• At $x = 1$: $f’'(1) = 6 > 0$, so there is a local minimum. The function value is $f(1) = 1 - 3 = -2$.

You can confirm this by graphing or by checking the first derivative’s sign on either side. This kind of classification problem is a staple of single-variable calculus, and it appears in course materials from many universities, including MIT OpenCourseWare.

Why the Test Is Useful, and When It Isn’t

The appeal of this method is efficiency. Once you can differentiate twice, you get an instant classification at each flat point without tracking signs across intervals. For polynomials, rational functions, and trigonometric functions, that’s usually the fastest route.

Its two main limits are worth knowing. First, it only works at flat points: if $f’©$ is undefined — a sharp corner like the one in $f(x) = |x|$ at $x = 0$ — the test doesn’t apply at all. Second, the zero case is genuinely ambiguous. For $f(x) = x^4$ at $x = 0$, we have $f’(0) = 0$ and $f’'(0) = 0$, yet there is a local minimum. For $f(x) = x^3$ at $x = 0$, the same derivative values occur but there is no extremum. Identical test output, different answers — hence the need to dig deeper. For multivariable functions, the idea generalizes using the Hessian matrix and its determinant, a topic usually covered in multivariable or vector calculus courses. Tutorials on building intuition for the single-variable case are widely available, including through Khan Academy.

A practical habit when choosing between the two tests: try the second derivative first, since it needs only one evaluation per point. If it returns zero or the derivative is undefined, switch immediately to the first derivative test rather than pushing a method past its limits.

Key Takeaways

• The second derivative test classifies critical points where $f’© = 0$: $f’‘© > 0$ means a local minimum, $f’'© < 0$ means a local maximum.
• If $f’‘© = 0$, the test is inconclusive — fall back to the first derivative test and check whether $f’$ changes sign.
• The test applies only at flat points where the first derivative exists; corners and cusps require other methods.
• The method works because the sign of the second derivative tells you whether the curve is concave up or concave down at the flat point.
• Always verify borderline or ambiguous cases by examining the function or its first derivative directly rather than assuming.