Critical Points: Step-by-Step Guide
A critical point of f is a value x = c in the domain of f where either f'(c) = 0 or f'(c) does not exist. Critical points are the candidates for local maxima, local minima, and points of horizontal tangency, and they are the foundation of optimization and curve sketching.
Math Racer skill guide · Calculus 1 · AP Calc AB / College
How to solve it, step by step
- Compute the derivative f'(x). Differentiate the function using whatever rules apply (power, product, quotient, chain).\[f'(x) = \frac{d}{dx}[f(x)]\]
- Set f'(x) = 0 and solve. Find every x in the domain where the derivative equals zero. These are stationary critical points.\[f'(x) = 0 \implies x = c_1, c_2, \dots\]
- Find where f'(x) is undefined but f(x) is defined. Identify any x in the domain of f where f' fails to exist (cusps, corners, vertical tangents). These are also critical points.\[f'(x) \text{ undefined while } f(x) \text{ defined}\]
- Classify each critical point. Use the first derivative test (sign change of f') or the second derivative test (sign of f'' at the point) to label each critical point as a local max, local min, or neither.\[f''(c) > 0 \Rightarrow \text{local min}; \;\; f''(c) < 0 \Rightarrow \text{local max}\]
Worked example
Find and classify all critical points of f(x) = x^3 - 3x^2 + 2.
Step 1: Differentiate: f'(x) = 3x^2 - 6x.
Step 2: Set f'(x) = 0: 3x^2 - 6x = 0, so 3x(x - 2) = 0, giving x = 0 and x = 2.
Step 3: f' is a polynomial, defined everywhere, so there are no additional critical points from undefined derivatives.
Step 4: Compute f''(x) = 6x - 6. At x = 0, f''(0) = -6 < 0, so x = 0 is a local maximum. At x = 2, f''(2) = 6 > 0, so x = 2 is a local minimum.
Step 5: Function values: f(0) = 2 (local max) and f(2) = 8 - 12 + 2 = -2 (local min).
Verification: Sample f(-0.1) = -0.001 - 0.03 + 2 = 1.969 < 2 = f(0), confirming local max. f(2.1) = 9.261 - 13.23 + 2 = -1.969 > -2 = f(2), confirming local min.
Tips and common mistakes
- Critical points must lie within the domain of f — values where f itself is undefined do not count.
- Always include points where f' is undefined (cusps, corners), not just where f' = 0.
- The first derivative test: f' changes from + to - means local max; from - to + means local min; no change means neither.
- The second derivative test fails if f''(c) = 0 — fall back to the first derivative test.
- Endpoints of a closed interval are NOT critical points but they ARE candidates for absolute extrema.
- For optimization, evaluate f at every critical point (and endpoints) and compare values.
- Sketch a sign chart of f' to visualize increasing/decreasing intervals between critical points.
- A horizontal tangent occurs at every critical point where f' = 0 — useful for graphing.