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Math101learn.math101.caSecond Derivative Test
A rigorous, example-driven guide to second derivative test, including hypotheses, method choice, verification, and practice.
The central idea
Suppose $f'(c)=0$ and $f''$ exists near $c$. If $f''(c)>0$, then $f$ has a local minimum at $c$; if $f''(c)<0$, it has a local maximum. If $f''(c)=0$ or does not exist, the test is inconclusive, not proof of no extremum.
Definitions, hypotheses, and notation
The test is a sufficient classification rule, not a universal one. At $f(x)=x^4$, both $f'(0)$ and $f''(0)$ vanish, yet the point is a strict local minimum. Higher-order terms or a first-derivative sign chart reveal what the zero second derivative cannot. Similarly, points where $f'$ is undefined must be classified by another method.
For a twice-differentiable function, the sign of $f''(c)$ describes how slopes change through a stationary point. A negative value means slopes are decreasing and cross from positive to negative nearby; a positive value means the reverse. The theorem packages that local sign behavior, while closed-interval absolute extrema still require endpoint comparison.
Conceptual meaning
At a stationary point, positive second derivative means the graph bends upward like a cup, placing the point locally low. Negative second derivative means it bends downward like a cap. The first-derivative condition is indispensable.
A dependable method and decision rule
- Find critical numbers from $f'=0$ or undefined within the domain.
- Use this test only at critical numbers where $f'(c)=0$ and $f''(c)$ can be evaluated.
- Compute $f''(c)$ and classify by its sign.
- When the result is zero or undefined, switch to the First Derivative Test.
- Report the point $(c,f(c))$ and whether the extremum is local.
Fully worked example
Graphical or geometric meaning
The graph is concave down around the maximum and concave up around the minimum. Equivalently, $f'$ is decreasing through zero at the maximum and increasing through zero at the minimum.
Common mistakes and why they fail
Verification and reasonableness checks
- Confirm $f'(c)=0$ before applying the test.
- Use the First Derivative Test as an independent sign-change check.
- Evaluate the original function to report coordinates.
Know when the test is inconclusive
At a critical point $c$ with $f'(c)=0$, a positive $f''(c)$ gives a strict local minimum and a negative value gives a strict local maximum. If $f''(c)=0$ or is undefined, the test gives no conclusion. It does not rule out an extremum: $x^4$ has a minimum at zero, whereas $x^3$ has neither type there. Use a first-derivative sign chart or nearby function values in an inconclusive case. Verify that $c$ is in the domain and actually critical. For absolute extrema on a closed interval, also evaluate endpoints; local classification alone cannot settle the global comparison.
Practice
- Classify $x^2$ at $0$.
- Classify $-x^2$ at $0$.
- What does the test say for $x^4$ at $0$?
Answers and brief solutions
- Local minimum because $f''(0)=2>0$.
- Local maximum.
- It is inconclusive, although a minimum exists.
Connections and next steps
Explore the idea
Tangent and accumulation explorer
Change one quantity at a time and connect what moves to Second Derivative Test.
Try it yourself
Hints are part of learning. Open one whenever it makes the next step feel possible.
For f(x)=x⁴−4x², how does the second derivative test classify x=0?
- f′=4x³−8x, so f′(0)=0.
- f″=12x²−8, so f″(0)=−8<0.
- Therefore x=0 is a local maximum.
End of lesson
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