AP®MathsCollege BoardCalculus BCStudy GuidesUnit 5: Analytical Applications of DifferentiationGraphs of Functions & Their DerivativesCritical Points

Critical Points (College Board AP® Calculus BC): Study Guide

Written by: Jamie Wood

Reviewed by: Dan Finlay

Updated on 8 May 2026

Local extrema versus global extrema

What is the difference between local and global extrema?

The term extremum (plural extrema) refers to maximum and minimum points on the graph of a function
An extremum can be either local (relative) or global
- A global extremum is the maximum or minimum for the whole of a function's domain
- A local or relative extremum is the maximum or minimum within a specific part of a function's domain
We can say that $f (x)$ has a
- global maximum at $x = c$ if
  - $f (x) \leq f (c)$ for every $x$ in the domain
- local maximum at $x = c$ if
  - $f (x) \leq f (c)$ for every $x$ in some open interval around $x = c$
- global minimum at $x = c$ if
  - $f (x) \geq f (c)$ for every $x$ in the domain
- local minimum at $x = c$ if
  - $f (x) \geq f (c)$ for every $x$ in some open interval around $x = c$
Every global extremum will also be a local extremum
- However, not all local extrema are global extrema

Graph with x and y axes showing a curve with labeled points: "global maximum" at point a, "local maximum," "global minimum," and "local minimum" at point b. — **The graph of a quartic function with domain between a and b. Note that global extrema can occur at the endpoints, or within an interval.**

Critical points

What is a critical point?

A critical point is a point where the first derivative of a function is
- equal to zero,
- or does not exist
  - In the case where $f^{'} (a)$ does not exist, the function itself must still be defined at $x = a$ (i.e. $f (a)$ must exist) in order for $x = a$ to be a critical point
  - E.g. $y = \frac{1}{x}$ does not have a critical point at $x = 0$ , but $y = x^{\frac{2}{3}}$ does
All local extrema occur at critical points
- However, not all critical points are local extrema

What different types of critical points are there?

Local minimums and maximums

At local minimums and maximums, the derivative is most often equal to zero
- I.e. for a function $f$ , $f^{'} (x) = 0$ at a local minimum or maximum
- The tangent at these points is horizontal

Depending on the function, these can also be global extrema
- E.g. The graph of $y = {(x - 2)}^{2}$ has a local minimum at (2,0), which is also a global minimum

Points of inflection

A point of inflection is a point where a graph changes concavity
- You can read more about concavity in the 'Concavity of Functions' study guide
Points of inflection are not local extrema
Only points of inflection where the first derivative is zero are critical points
- It is possible for points of inflection to exist where the first derivative is not zero
A point of inflection with the first derivative zero is sometimes referred to as a saddle point
- E.g. $y = x^{3}$ has a saddle point at $x = 0$

Points where the derivative does not exist

As well as where the derivative is zero, a critical point can occur at points where the derivative does not exist
- If the derivative does not exist at a point, the function itself must still be defined at that point to be a critical point
Consider the function $f$ defined by $f (x) = \frac{1}{x}$ ,
- The function is undefined at $x = 0$ , so will not have a critical point at $x = 0$
Consider the function $g$ defined by $g (x) = \sqrt[3]{x} = x^{\frac{1}{3}}$
- The function is defined at $x = 0$ (it has a value of $g (0) = 0$ )
- It has a first derivative of $g^{'} (x) = \frac{1}{3} x^{- \frac{2}{3}} = \frac{1}{3 \sqrt[3]{x^{2}}}$
- The first derivative does not exist at $x = 0$
- So $g$ has a critical point at $x = 0$
The image below shows a similar example

Graph of f(x) = (x - 2)^(1/3) with annotations. f(x) defined at x=2, f(2)=0. Critical point at x=2. f'(x)=1/3(x-2)^(-2/3). f'(x) does not exist at x=2. — **An example of a critical point where the derivative is undefined**

It is also possible for a critical point to exist at a point that is defined but not continuous
- This means the derivative does not exist at that point
- The function jumps to a higher or lower value

Graph of a curve with a discontinuity 'hole' at x=a (where a>0), and a single filled-in point above the curve also at x=a — **An example of a local maximum point at a point which is not continuous**

Examiner Tips and Tricks

It is very unlikely that this type of critical point will appear in the exam. However, it is good for you to understand them.

Worked Example

Find the coordinates of the critical point(s) on the graph of the function $f$ defined by $f (x) = 4 x^{3} - 30 x^{2} + 48 x + 3$ .

Answer:

Find the points where the derivative is equal to zero

$f^{'} (x) = 12 x^{2} - 60 x + 48$

$\begin{array}{rcl} 12 x^{2} - 60 x + 48 & = & 0 \\ x^{2} - 5 x + 4 & = & 0 \\ (x - 4) (x - 1) & = & 0 \end{array}$

$x = 4$ or $x = 1$

$f^{'} (x)$ exists for all $x$ , so there are no other critical points (i.e. where the derivative does not exist)

Find the corresponding $y$ values for these two critical points

$f (1) = 25 f (4) = - 29$

A graph of the function is shown below

Graph of a cubic function with labeled points at (1, 25) and (4, -29). The curve increases, decreases, and then increases again with axes marked.

Critical points at (1, 25) and (4, -29)

Worked Example

Find the coordinates of the critical point(s) on the graph of the function $g$ defined by $g (x) = {(x - 3)}^{\frac{2}{3}}$ .

Answer:

Differentiate the function using the chain rule

$g^{'} (x) = \frac{2}{3} {(x - 3)}^{- \frac{1}{3}} = \frac{2}{3 {(x - 3)}^{\frac{1}{3}}}$

There are no points where the derivative is equal to zero

However the derivative will be undefined when $x = 3$ , as the denominator will be zero

The function is defined when $x = 3$ , as $g (3) = {(0)}^{\frac{2}{3}} = 0$

However $g^{'} (3)$ is undefined, so there is a critical point at $x = 3$

A graph of the function is shown below

Graph of the quadratic function y = (x - 3)^(2/3) on a grid, with a cusp at (3, 0).

Critical point at (3, 0)

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Critical Points (College Board AP® Calculus BC): Study Guide

Local extrema versus global extrema

What is the difference between local and global extrema?

Critical points

What is a critical point?

What different types of critical points are there?

Local minimums and maximums

Points of inflection

Points where the derivative does not exist

Examiner Tips and Tricks

Worked Example

Worked Example

Unlock more, it's free!

Join the 100,000+ Students that ❤️ Save My Exams

Author: Jamie Wood

Reviewer: Dan Finlay