The Second Derivative Test for Extrema
Simple Explanation
At a critical point c where f'(c)=0, the Second Derivative Test classifies it using f''(c): if f''(c)>0 the graph is concave up there, so c is a local minimum; if f''(c)<0 the graph is concave down there, so c is a local maximum; if f''(c)=0 the test is inconclusive and the First Derivative Test must be used instead.
Why Do We Need It?
It is often faster than checking the sign of f' on both sides of a critical point (the First Derivative Test) — a single evaluation of f'' at the critical point is usually enough.
Formula
The Second Derivative Test
At a critical point c where f'(c)=0: f''(c)>0 ⟹ local min; f''(c)<0 ⟹ local max; f''(c)=0 ⟹ inconclusive
An alternative (often faster) way to classify a critical point, using the sign of the second derivative there instead of checking the sign change of the first derivative on both sides.
- c
- — a critical point where f'(c)=0
When to use it: Whenever a critical point needs to be classified, and computing the second derivative is more convenient than the First Derivative Test.
Worked Example
Classify critical points using the Second Derivative Test
Classify the critical points of f(x) = x³ − 3x (at x=−1 and x=1) using the Second Derivative Test.
Why Does This Work?
If f''(c)>0, the graph is concave up right around c, which means the tangent line (already flat, since f'(c)=0) sits below the curve on both sides — exactly the shape of a local minimum. The mirror-image argument with f''(c)<0 gives a local maximum.
Real-Life Example
Confirming a minimum-cost production level
A cost function has a critical point at a certain production level, suspected to be the cheapest level to operate at.
Checking that the cost function's second derivative is positive there confirms it truly is a minimum (concave up), rather than a maximum or an inflection point.
Practice
At a critical point c, f''(c) = 5. What does the Second Derivative Test conclude?
MediumAt a critical point c, f''(c) = 0. What must be done to classify c?
HardCommon mistake
Applying the Second Derivative Test at a point that is not actually a critical point — the test only classifies points where f'(c)=0 in the first place.
Quick Review
- f''(c)>0 at a critical point c ⟹ local minimum. f''(c)<0 ⟹ local maximum.
- f''(c)=0 ⟹ inconclusive — fall back to the First Derivative Test.
- Usually faster than the First Derivative Test, but not always applicable.