A critical point is a value in the domain of a function where the derivative equals zero or is undefined. On the AP Calculus exam, critical points are the candidate locations for relative (local) maxima and minima, and the starting step for the First Derivative, Second Derivative, and Candidates Tests.
A critical point of a function f is an x-value in the domain of f where f'(x) = 0 or where f'(x) does not exist. Think of it as a place where the function's instantaneous rate of change either flattens out (a horizontal tangent line) or breaks down (a corner, cusp, or vertical tangent). Either way, the function might be switching from increasing to decreasing or vice versa right there.
The word that matters is candidate. A critical point is a possible location of a relative max or min, not a guaranteed one. The classic counterexample is f(x) = x³, which has f'(0) = 0 but no extremum at x = 0 because the function never stops increasing. Per the CED, the first derivative determines the location of relative extrema (FUN-4.A.2), and absolute extrema on a closed interval can only occur at critical points or endpoints. So finding critical points is step one of nearly every analysis problem in Unit 5, but you always need a test (First Derivative, Second Derivative, or Candidates) to say what kind of point it actually is.
Critical points are the backbone of Unit 5 (Analytical Applications of Differentiation). They support learning objectives AP Calc 5.4.A, 5.5.A, and 5.7.A, which all ask you to justify conclusions about a function's behavior using its derivatives. The First Derivative Test (Topic 5.4) checks the sign of f' on each side of a critical point. The Second Derivative Test (Topic 5.7) plugs the critical point into f''. The Candidates Test (Topic 5.5) evaluates f at every critical point plus the endpoints to find absolute extrema. Optimization problems (Topics 5.10-5.11, objectives AP Calc 5.10.A and 5.11.A) are the same idea in a word-problem costume. You model a quantity, take a derivative, set it to zero, and interpret the critical point in context. Critical points also echo in Unit 4, where interpreting f'(x) = 0 in context (AP Calc 4.1.A) means the quantity is momentarily not changing.
Keep studying AP Calculus Unit 5
Visual cheatsheet
view galleryFirst Derivative Test (Unit 5)
This is the most direct companion to critical points. Once you've found where f' = 0 or is undefined, you check the sign of f' on each side. A switch from positive to negative means a relative max, negative to positive means a relative min, and no sign change means no extremum at all.
Candidates Test (Unit 5)
On a closed interval, absolute extrema can only live at critical points or endpoints. The Candidates Test takes that fact literally. You list every critical point plus both endpoints, evaluate f at each, and the biggest and smallest outputs win.
Optimization Problems (Unit 5)
Every optimization problem secretly ends in a critical point hunt. Whether you're minimizing material for a box or maximizing profit, you build a function, differentiate it, set the derivative to zero, and then justify that your critical point really is the max or min the problem wants.
Logistic Models (Unit 7)
The 'derivative equals zero' idea shows up again with dy/dt = ky(a − y). Setting the rate to zero finds equilibrium values, and the population grows fastest at half the carrying capacity, which you find by maximizing dy/dt. It's the same critical-point logic applied to a differential equation.
Critical points are everywhere in both MCQs and FRQs. Multiple-choice stems ask you to identify critical points from a formula, a graph of f', or a table, and then classify them. The classic FRQ setup gives you the graph of f' (not f) and asks where f has a relative max or min. The 2017, 2022, and 2023 exams all ran this exact play with a graph of f' made of line segments and a semicircle. You read off where f' crosses zero, check the sign change, and write a justification like 'f has a relative maximum at x = c because f' changes from positive to negative at x = c.' The 2024 BC exam flagged a critical point by telling you the graph has a horizontal tangent at x = −2. The justification sentence is where points are won or lost. Saying 'f'(c) = 0' alone earns nothing; you must cite the sign behavior of f' or the sign of f''(c).
Critical points come from the FIRST derivative (f' = 0 or undefined) and are candidates for maxima and minima. Inflection points come from the SECOND derivative and mark where concavity changes. A point can be both (like x = 0 for some functions), but they answer different questions. On the graph of f', critical points of f are where f' crosses zero, while inflection points of f are where f' has a max or min. Mixing these up on a 'here's the graph of f'' FRQ is one of the most common ways to lose points.
A critical point is an x-value in the domain of f where f'(x) = 0 or f'(x) is undefined.
Critical points are candidates for relative extrema, not guarantees; f(x) = x³ has a critical point at x = 0 with no max or min there.
Absolute extrema on a closed interval can only occur at critical points or at the endpoints, which is exactly why the Candidates Test works.
To classify a critical point, use the First Derivative Test (sign change in f') or the Second Derivative Test (sign of f'' at the point), and your written justification must cite that evidence.
If a continuous function has exactly one critical point on an interval and it's a relative extremum, the CED says it's also the absolute extremum on that interval.
When you're given the graph of f', the critical points of f are where that graph touches or crosses the x-axis.
A critical point is a value in the domain of a function where the derivative is zero or undefined. It's the candidate location for a relative maximum or minimum, and finding critical points is the first step in the First Derivative Test, Second Derivative Test, and Candidates Test.
No. A critical point is only a candidate. For f(x) = x³, f'(0) = 0 but x = 0 is neither a max nor a min because f' doesn't change sign there. You always need a test to classify a critical point.
Critical points come from the first derivative (f' = 0 or undefined) and flag possible extrema. Inflection points come from the second derivative and flag where concavity changes. On a graph of f', critical points of f are where f' hits zero, while inflection points of f are where f' peaks or bottoms out.
Yes, as long as the point is in the function's domain. Corners (like f(x) = |x| at x = 0), cusps, and vertical tangents all create critical points even though there's no horizontal tangent line there.
Look for x-values where the graph of f' touches or crosses the x-axis, since those are where f'(x) = 0. This is the standard setup in released FRQs, including 2017, 2022, and 2023, where the graph of f' is built from line segments and a semicircle and you classify each zero by whether f' changes sign.