Critical Point — Definition, Formula & Examples

Critical Point

A point (x, y) on the graph of a function at which the derivative is either 0 or undefined. A critical point will often be a minimum or maximum, but it may be neither.

Note: Finding critical points is an important step in the process of curve sketching.

See also

Second order critical point

Key Formula

f'(c) = 0 \quad \text{or} \quad f'(c) \text{ is undefined}

Where:

$f'(c)$ = The derivative of the function f evaluated at x = c
$c$ = The x-value where the critical point occurs (must be in the domain of f)

Worked Example

Problem: Find all critical points of f(x) = x³ − 6x² + 9x + 2.

Step 1: Find the derivative of f(x) using the power rule.

f'(x) = 3x^2 - 12x + 9

Step 2: Set the derivative equal to zero and solve for x.

3x^2 - 12x + 9 = 0

Step 3: Factor out the common factor of 3, then factor the quadratic.

3(x^2 - 4x + 3) = 0 \implies 3(x - 1)(x - 3) = 0

Step 4: Solve for x to get the critical values.

x = 1 \quad \text{or} \quad x = 3

Step 5: Find the y-coordinates by substituting back into f(x). At x = 1: f(1) = 1 − 6 + 9 + 2 = 6. At x = 3: f(3) = 27 − 54 + 27 + 2 = 2.

\text{Critical points: } (1, 6) \text{ and } (3, 2)

Answer: The critical points are (1, 6) and (3, 2).

Another Example

This example shows a critical point where the derivative is undefined rather than zero. The graph of x^{2/3} has a cusp at the origin — a sharp corner where no tangent line exists. This is a local minimum even though f'(0) does not equal zero.

Problem: Find all critical points of f(x) = x^{2/3}.

Step 1: Find the derivative using the power rule.

f'(x) = \frac{2}{3}x^{-1/3} = \frac{2}{3\sqrt[3]{x}}

Step 2: Check where f'(x) = 0. The numerator is 2, which is never zero, so f'(x) never equals zero.

\frac{2}{3\sqrt[3]{x}} \neq 0 \text{ for any } x

Step 3: Check where f'(x) is undefined. The derivative has a denominator of 3∛x, which equals zero when x = 0.

f'(0) \text{ is undefined}

Step 4: Verify that x = 0 is in the domain of f. Since f(0) = 0^{2/3} = 0, the point (0, 0) is on the graph.

\text{Critical point: } (0, 0)

Answer: The only critical point is (0, 0), where the derivative is undefined.

Frequently Asked Questions

What is the difference between a critical point and an inflection point?

A critical point occurs where f'(x) = 0 or f'(x) is undefined, making it a candidate for a local max or min. An inflection point occurs where the concavity of f changes, meaning f''(x) changes sign. A point can be both: for example, if f'(c) = 0 but the point is neither a max nor a min, and f''(c) changes sign there, it is both a critical point and an inflection point.

Is every critical point a maximum or minimum?

No. A critical point can be a local maximum, a local minimum, or neither. For example, f(x) = x³ has a critical point at x = 0 (since f'(0) = 0), but this point is neither a max nor a min — it is an inflection point. You need a test like the First Derivative Test or the Second Derivative Test to classify a critical point.

How do you tell if a critical point is a max or min?

Use the First Derivative Test: check the sign of f'(x) on either side of the critical point. If f' changes from positive to negative, you have a local max; from negative to positive gives a local min. Alternatively, use the Second Derivative Test: if f''(c) > 0 the point is a local min, and if f''(c) < 0 it is a local max. If f''(c) = 0, the test is inconclusive.

Critical Point vs. Inflection Point

	Critical Point	Inflection Point
Definition	Where f'(x) = 0 or f'(x) is undefined	Where the concavity of f changes (f'' changes sign)
Derivative condition	Involves the first derivative f'(x)	Involves the second derivative f''(x)
What it identifies	Candidates for local maxima or minima	Points where the curve changes from concave up to concave down (or vice versa)
Example on f(x) = x³	x = 0 (since f'(0) = 0)	x = 0 (since f'' changes sign from negative to positive)

Why It Matters

Critical points are central to calculus-based optimization — finding the maximum profit, minimum cost, or shortest distance in applied problems all start with finding critical points. In AP Calculus and college courses, nearly every curve-sketching problem requires you to locate critical points first, then classify them. Understanding critical points also builds the foundation for multivariable calculus, where the concept extends to partial derivatives equaling zero.

Common Mistakes

Mistake: Assuming every critical point is a local max or min.

Correction: A critical point can be neither. For f(x) = x³, the point (0, 0) has f'(0) = 0 but is not a max or min. Always use the First or Second Derivative Test to classify critical points.

Mistake: Forgetting to check where the derivative is undefined.

Correction: Critical points include x-values where f'(x) does not exist (as long as f(x) itself is defined there). For instance, f(x) = |x| has a critical point at x = 0 because f'(0) is undefined, and this turns out to be a minimum.

Related Terms

Derivative — Critical points are defined using the derivative
Maximum of a Function — Local maxima occur at certain critical points
Minimum of a Function — Local minima occur at certain critical points
Curve Sketching — Uses critical points to determine graph shape
Second Order Critical Point — Where the second derivative is zero or undefined
Function — Critical points are defined for functions
Graph of an Equation or Inequality — Critical points are visible features on graphs
Point — A critical point is a specific point (x, y)