A critical point of a function is an input x = c in its domain where the derivative is 0 or does not exist. These are the only places a smooth graph can turn around, so every local maximum or minimum happens at a critical point. For f(x) = x2 − 4x, f′(x) = 2x − 4 = 0 at x = 2.
In Spanish
punto crítico
Also called
critical number, critical value, stationary point
Grade level
Calculus
CalculusAP
Critical Point in a picture
Both turning points of x3 − 3x are critical points, where the tangent line is flat. The sign of f′ changes from + to − at the maximum and from − to + at the minimum.
Formal definition and notation
A numberc in the interior of the domain of f is a critical number (critical point) of f if f′(c) = 0 or f′(c) is undefined. Fermat’s theorem states that if f has a local extremum at c and f is differentiable at c, then f′(c) = 0. The converse is false: a critical point need not be an extremum. Some books call the point (c, f(c)) on the graph the critical point.
Symbol
Read it as
Meaning
f′(c) = 0
f prime of c equals zero
horizontal tangent at x = c
f′(c) undefined
f prime of c does not exist
a corner, cusp or vertical tangent at x = c
(c, f(c))
c, f of c
the point on the graph at a critical number
Key facts
Local maxima and minima of a function can only occur at critical points (or at endpoints of a closed interval).
Not every critical point is a max or min: f(x) = x3 has f′(0) = 0 but keeps rising through 0.
Where f′ does not exist counts too: |x| has a corner and x23 has a cusp at 0.
A value where f itself is undefined is not a critical point, even if f′ is undefined there.
To find the absolute max or min on [a, b], compare f at the critical points and the two endpoints.
How to find critical points
How to find critical points
f′(x) = 0 or f′(x) undefined, with x in the domain
f′(x)
the derivative of the function
x
a candidate; keep it only if f(x) is defined there
Classify each one with the first derivative test: if f′ changes from + to −, it is a local maximum; from − to +, a local minimum; no sign change, neither.
Critical Point examples, step by step
Example 1: A quadratic
Warm-up
Find the critical point of f(x) = x2 − 8x + 3.
Differentiate: f′(x) = 2x − 8.
Set it to 0: 2x − 8 = 0, so x = 4.
f′ exists everywhere, so there are no other critical points.
Answer: The only critical point is x = 4 (the vertex of the parabola, a minimum).
Example 2: Find and classify
Core
Find and classify the critical points of f(x) = x3 − 12x + 1.
f′(x) = 3x2 − 12 = 3(x − 2)(x + 2), which is 0 at x = −2 and x = 2.
Signs of f′: positive for x < −2, negative between, positive for x > 2.
At x = −2, + to −: local maximum, f(−2) = −8 + 24 + 1 = 17.
At x = 2, − to +: local minimum, f(2) = 8 − 24 + 1 = −15.
Answer: Local maximum at (-2, 17) and local minimum at (2, -15).
It is undefined at x = 0, and f(0) = 0 is defined, so x = 0 is a critical point.
The graph has a sharp cusp there, and it is a minimum.
Answer: The only critical point is x = 0, where the derivative does not exist.
Common mistakes
✗ Mistakef′(0) = 0 for f(x) = x3, so x = 0 is a maximum or minimum.
✓ Insteadx = 0 is a critical point, but neither a max nor a min.
f′(x) = 3x2 is positive on both sides of 0, so the graph keeps rising. A critical point is only a candidate for an extremum.
✗ MistakeOnly solving f′(x) = 0 for f(x) = |x − 3|, and finding nothing.
✓ Insteadx = 3 is a critical point because f′(3) does not exist.
Critical points include places where the derivative is undefined, such as corners and cusps.
✗ Mistakex = 0 is a critical point of f(x) = 1x, since f′(0) is undefined.
✓ Instead1x has no critical points.
A critical point must be in the domain of f. f(0) itself does not exist, so 0 cannot be a critical point.
Not to be confused with
Critical Pointvs.Inflection point
An inflection point is where the concavity changes, found from f′′. A critical point comes from f′. For x3 − 3x, the critical points are ± 1 and the inflection point is 0.
Critical Pointvs.Zero of a function
A zero is where f(x) = 0, so the graph crosses the x-axis. A critical point is where f′(x) = 0 or is undefined, so the graph is flat or sharp. x2 − 4 has zeros ± 2 but its critical point is 0.
The vertex is the one critical point of a parabola. Other functions can have many critical points, or none.
Quick check: 3 questions
Question 1 of 3
What are the critical points of f(x) = x3 − 3x2?
Show answer and explanation
Answer: C.f′(x) = 3x2 − 6x = 3x(x − 2), which is 0 at 0 and 2. x = 0 and 3 are the zeros of f itself, and x = 1 is the inflection point.
Question 2 of 3
If f′ changes from negative to positive at x = c, then f has
Show answer and explanation
Answer: B. Falling and then rising makes a valley, so f has a local minimum at c. Positive to negative would be a maximum. An inflection point depends on f′′, not on the sign of f′.
Question 3 of 3
Which statement is true?
Show answer and explanation
Answer: C. A critical point is an input in the domain where f′ is 0 or undefined. x3 shows a critical point that is not an extremum; corners show the undefined case; zeros of f are unrelated.
Critical Point: frequently asked questions
How do you find critical points?
Find the derivative f′(x). Then solve f′(x) = 0, and also find every x where f′(x) is undefined, such as where its denominator is 0. Keep only the values that are in the domain of f. For f(x) = x4 − 2x2, f′(x) = 4x3 − 4x = 4x(x − 1)(x + 1), so the critical points are −1, 0 and 1.
Is every critical point a maximum or a minimum?
No. A critical point is only a candidate. Check the sign of f′ on each side: + to − means a local maximum, − to + means a local minimum, and no change means neither. The classic counterexample is f(x) = x3 at 0: the slope is 0 there, but the graph flattens for a moment and keeps rising.
What is the difference between a critical point and a critical value?
Usage varies between books. Often the critical number or critical point is the input c, and the critical value is the output f(c). Other books call the whole point (c, f(c)) the critical point. In f(x) = x2 − 8x + 3, the critical number is 4 and the function value there is −13.
Updated November 2026·Reviewed by the Effortless Math team·Standards: n/a
Cite this entry
Effortless Math Team. “Critical Point.” Effortless Math Dictionary. Updated November 2026. https://www.effortlessmath.com/math-dictionary/critical-point/
Flashcard
TermCritical Pointf′(c) = 0 or undefined
DefinitionAn input where the derivative is 0 or does not exist, the only places a graph can turn. x2 − 4x has a critical point at x = 2.
DerivativeA derivative measures how fast a function’s output changes at one exact input: its instantaneous rate of change. On a graph it is the slope of the tangent line at that point. The derivative of f(x) = x2 is f′(x) = 2x, so at x = 3 the curve is rising with slope 6.Calculus · CalculusNumberA number is an amount or value used to count, measure, order and compare. We write numbers with digits (7), words (seven) or other symbols, but the number itself is the amount: 7 dots, the word seven and the numeral 7 all show the same number. Numbers include whole numbers, fractions, decimals and negatives.Number System · Kindergarten, Grade 1, Grade 2DomainThe domain of a function is the set of all input values (usually x-values) the function can accept. Unless a problem says otherwise, it is every real number except values that would divide by zero or take an even root of a negative number. On a graph, the domain is how far the graph reaches left and right.Functions · Grade 8, Algebra 1, Algebra 2UndefinedUndefined describes an expression that has no value, most often because it divides by zero. 70 is undefined, since no number times 0 gives 7. A vertical line has an undefined slope, and 3x − 2 is undefined at x = 2. Undefined does not mean zero: 07 = 0, but 70 has no value.Problem Solving · Grade 6, Grade 8, Algebra 1ParabolaA parabola is the U-shaped curve that is the graph of a quadratic function such as y = x2. It is symmetric about a line called the axis of symmetry, and its turning point is the vertex. Geometrically, it is every point equally far from a fixed point (the focus) and a fixed line (the directrix).Functions · Algebra 1, Algebra 2, GeometryVertex of a ParabolaThe vertex of a parabola is its turning point: the lowest point when the parabola opens up, or the highest point when it opens down. 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A triangle has 3 sides and a square has 4, and adding the lengths of all the sides gives the perimeter.Geometry · Kindergarten, Grade 1, Grade 2X-AxisThe x-axis is the horizontal number line on the coordinate plane. It crosses the vertical y-axis at the origin, (0, 0). The x-coordinate of a point tells how far to move along the x-axis, and every point on the x-axis has a y-coordinate of 0, such as (5, 0).Geometry · Grade 5, Grade 6, Grade 7DenominatorThe denominator is the bottom number in a fraction. It tells how many equal parts make one whole: in 35, the 5 means the whole is cut into 5 equal parts, and the numerator 3 counts the parts you have. The denominator can never be 0.Number System · Grade 3, Grade 4, Grade 5CounterexampleA counterexample is a single example that shows a general statement is false. It fits all the conditions of the statement but does not have the claimed result. To disprove ‘every prime number is odd,’ one counterexample is enough: 2 is prime and even. Examples that agree with a claim never prove it, but one counterexample disproves it.Geometry · Grade 8, Algebra 1, GeometrySlopeSlope is a number that measures how steep a line is: the change in y divided by the change in x between any two points on the line, often called rise over run. A positive slope rises from left to right, a negative slope falls, and a horizontal line has slope 0.Algebra · Grade 8, Algebra 1Chain RuleThe chain rule is the rule for the derivative of a composite function: if y = f(g(x)), then y′ = f′(g(x)) · g′(x). Differentiate the outer function, leave the inner function inside, then multiply by the derivative of the inner function. For (x2 + 1)3, the derivative is 3(x2 + 1)2 · 2x.Calculus · CalculusLimitA limit is the value a function gets closer and closer to as the input approaches a certain number. We write limx → af(x) = L. The function need not even be defined at a. For f(x) = x2 − 1x − 1, f(1) is undefined, but the limit as x → 1 is 2.Calculus · Precalculus, CalculusIntegralAn integral adds up infinitely many tiny pieces to find a total, such as the area under a curve. The definite integral ∫baf(x) dx is the signed area between the graph of f and the x-axis from x = a to x = b. An indefinite integral is an antiderivative: ∫ 2xdx = x2 + C.Calculus · CalculusCalculusCalculus is the branch of mathematics that studies how quantities change. Its two main tools are the derivative, which measures the rate of change at an instant (the slope of a curve at a point), and the integral, which adds up infinitely many tiny pieces (the area under a curve). Both are built on limits.Calculus · CalculusEnd BehaviorEnd behavior describes what the y-values of a graph do at the far left and far right, as x → −∞ and x → ∞. For a polynomial, only the leading term matters: its degree (even or odd) and the sign of its coefficient. For example, f(x) = −2x3 + x rises on the left and falls on the right.Functions · Algebra 2, Precalculus