Second Derivative Test
When analyzing the graph of a function, one of the most important tasks is identifying and classifying its critical points — the places where the derivative equals zero or is undefined. While the First Derivative Test examines the sign change of f′(x) around a critical point, the Second Derivative Test offers a faster and often more elegant approach: it uses the concavity of the curve at that point to determine whether the critical point is a local minimum, local maximum, or neither. The key idea is rooted in geometry. If a function is concave up at a critical point (shaped like a bowl opening upward), the point must be sitting at the bottom of that bowl, making it a local minimum. Conversely, if the function is concave down (shaped like a cap opening downward), the critical point sits at the top, making it a local maximum. Concavity is precisely what the second derivative f″(x) measures: positive f″ means concave up, and negative f″ means concave down. The Second Derivative Test is a standard tool in calculus-based curve analysis, appearing in optimization problems, physics (finding equilibrium positions), and economics (maximizing profit or minimizing cost). It works hand-in-hand with concepts such as inflection points and intervals of concavity, forming a complete picture of how a function behaves across its domain.
- Find the first derivative
- Solve for critical points
- Find the second derivative
- Evaluate f″ at each critical point
- State the conclusion
Find the first derivative
Compute f′(x). This expression gives the slope of the tangent line at every point and will be used to locate critical points.
Solve for critical points
Set f′(x) = 0 and solve for x. Any value c where f′(c) = 0 (and f is defined there) is a critical point and a candidate for a local extremum.
Find the second derivative
Compute f″(x) by differentiating f′(x). This measures the rate of change of the slope and tells us the concavity of f.
Evaluate f″ at each critical point
Substitute each critical point c into f″(x): If f″(c) > 0, the graph is concave up at c, so f has a local minimum there. If f″(c) < 0, the graph is concave down at c, so f has a local maximum there. If f″(c) = 0, the test is inconclusive — the point could be a local min, local max, or an inflection point; you must use another method such as the First Derivative Test.
State the conclusion
Report the nature of each critical point (local maximum or local minimum) along with its coordinates (c, f(c)). If the test was inconclusive, explicitly note that and switch to the First Derivative Test.
The Key Equation
- f'(c) — First derivative at c (dimensionless (slope)): The slope of the tangent line to f at x = c; equals zero at every critical point tested.
- f''(c) — Second derivative at c (dimensionless (curvature)): The rate of change of the slope at x = c; its sign encodes whether the curve bends upward or downward at c.
- c — Critical point (input value (x-units)): A value in the domain of f where f′(c) = 0, making it a candidate for a local maximum or minimum.
Worked Example
Problem. Find and classify all local extrema of f(x) = x³ − 3x² − 9x + 5 using the Second Derivative Test.
Answer. Local maximum at (−1, 10) and local minimum at (3, −22).
Origins · 1 minGottfried Wilhelm Leibniz and how this was discovered
Leibniz independently developed calculus alongside Isaac Newton, introducing the notation d/dx and d²y/dx² that we still use today for derivatives and second derivatives. His systematic notation made it far easier to express and manipulate higher-order derivatives, laying the foundation for tools like the Second Derivative Test. Leibniz also designed one of the first mechanical calculators, the Stepped Reckoner, which could multiply and divide — a remarkable feat of engineering in the 1670s.
Related Concepts
- First Derivative Test — The First Derivative Test is the alternative method for classifying critical points; it is essential when the Second Derivative Test gives an inconclusive result (f″(c) = 0).
- Concavity and Inflection Points — Understanding concavity — what f″(x) > 0 and f″(x) < 0 mean geometrically — is the conceptual foundation the Second Derivative Test rests upon.
- Critical Points — The Second Derivative Test only applies at critical points, so finding them by setting f′(x) = 0 is an essential prerequisite step.
- Optimization Problems — The Second Derivative Test is a primary tool in applied optimization — confirming whether a critical point yields the maximum or minimum value of a quantity you want to optimize.