Critical points of a function
Also known as: critical numbers, critical values
Critical points of a function are the points in its domain where the derivative equals zero or does not exist. They are the only candidates for local maximums and minimums, making them the starting point for optimization problems.
A critical point of a function f is a value x = c in the domain of f where f′(c) = 0 or f′(c) does not exist. Geometrically, these are the places where the graph has a horizontal tangent line (derivative zero) or a sharp corner, cusp, or vertical tangent (derivative undefined). Both conditions matter — the absolute value function f(x) = |x| has a critical point at x = 0 even though its derivative is never zero.
To find critical points, take the derivative, set it equal to zero, and solve; then separately check where the derivative is undefined while the function itself is still defined. For example, f(x) = x³ − 3x has derivative f′(x) = 3x² − 3, which equals zero at x = 1 and x = −1, so those are the critical points.
Critical points matter because of a key theorem: any local maximum or minimum of a function must occur at a critical point. The converse is not true — x = 0 is a critical point of f(x) = x³, but it is neither a max nor a min. To classify a critical point, apply the first derivative test (check whether f′ changes sign from positive to negative for a max, or negative to positive for a min) or the second derivative test.
The AP Calculus AB exam tests critical points heavily: finding them from a formula or a graph of f′, classifying them with the first derivative test, and using them alongside endpoints to locate absolute extrema on a closed interval. They also anchor optimization and curve-sketching problems.
Key takeaways
- A critical point occurs where f′(x) = 0 or f′(x) is undefined, provided x is in the domain of f.
- Every local maximum or minimum occurs at a critical point, but not every critical point is an extremum.
- Classify critical points with the first derivative test (sign change of f′) or the second derivative test.
- For absolute extrema on a closed interval, compare function values at critical points and endpoints.
- AP Calculus AB tests finding and classifying critical points in both algebraic and graphical settings.