What are critical points? In calculus, a critical point is a point on a function where the derivative of the function is either zero or undefined.

We say that $x = c$ is a critical number of the function $f$ if either $f′(c) = 0$, or $f′(c)$ is undefined. We say that $(c, f(c))$ are the critical points of the function.

The critical point definition is sometimes confused with the definition of critical numbers. The critical numbers of a function $f$ are the x-values $c$ in the domain of the function for which $f’(c) = 0$ or $f’(c)$ is undefined. The critical points of a function are the points on a graph whose coordinates are $(c, f(c))$.

Critical points are a big deal because they can help us identify relative extrema, like relative minima and maxima. These are the peaks and valleys on a graph.

All local extrema occur at critical points, but not all critical points are local extrema. To determine which critical points are local minima or maxima, we use the First Derivative Test or Second Derivative Test.

When thinking about critical points note that functions can have both increasing and decreasing intervals.

Suppose a function f is differentiable on the interval $I$. Then:

If $f’(x) > 0$ for each $x$ on $I$, then $f$ is increasing on $I$.

If f’(x) < 0 for each x on I, then f is decreasing on I.

If $f’(x) = 0$ for each $x$ on $I$, then $f$ is constant on $I$.

Critical points that are relative extrema occur at the points where the curve changes direction from increasing to decreasing, or decreasing to increasing. In other words, the first derivative $f’(x)$ can only change sign from positive to negative, or negative to positive, by crossing points where $f’(x) = 0$ or points where $f’(x)$ does not exist.

3 Types of Critical Points

Three types of critical points exist in single-variable calculus:

Relative Maxima

For a function $f$, relative maxima occur at the critical points $(c, f(c))$ where both of the following criteria are satisfied:

$f’(c) = 0$ or $f’(c)$ does not exist

f’(x) > 0 on the left of x = c, and f’(x) < 0 on the right of x = c

Relative Minima

For a function $f$, relative minima occur at the critical points $(c, f(c))$ where both of the following criteria are satisfied:

$f’(c) = 0$ or $f’(c)$ does not exist

f’(x) < 0 on the left of x = c, and f’(x) > 0 on the right of x = c

Vertical Tangents

Some critical points are neither relative maxima nor relative minima. At vertical tangents, the derivative is undefined. These are inflection points, which are the points where the curve changes concavity from concave up to concave down, or concave down to concave up.

Take the second derivative on either side of the critical point to determine if the sign of the second derivative changes from positive to negative or negative to positive at this point. If it does, then the critical point is an inflection point. Note: not all inflection points are critical points!

How To Find Critical Points

In this section, we will learn how to find critical numbers, how to find critical points, and how to classify them.

Take the first derivative of the function.

Set the first derivative function equal to zero and solve for $x$. Also solve for any points where the first derivative function is undefined. These x-values $c$ are the critical numbers of the function.

Plug $c$ into $f(x)$ to obtain the y-values. The coordinates $(c, f(c))$ are the critical points of the function.

To determine if the critical point is a relative maxima, relative minima, or neither, you can use the First Derivative Test. This test involves analyzing the sign of the first derivative of the function on either side of the critical point.

If the sign of the first derivative changes from positive to negative at the critical point, it's a relative maximum. The function changes from increasing to decreasing at this point.

If the sign of the first derivative changes from negative to positive at the critical point, it's a relative minimum. The function changes from decreasing to increasing at this point.

You can also use the Second Derivative Test to determine if the critical point is a local maximum or local minimum. Find each stationary point of the function, which are the critical points $c$ where $f’(c) = 0$. For each stationary point $c$:

If $f"(c)$ is positive, then the function has a relative minimum at $c$.

If $f"(c)$ is negative, then the function has a relative maximum at $c$.

If $f"(c)$ is zero or non-existent, then the test is inconclusive.

To determine if a critical point is an inflection point, use test points to establish if the second derivative $f’’(x)$ changes sign from positive to negative, or negative to positive, on either side of $x=c$.

If f”(x) > 0 on the left of x = c, and f”(x) < 0 on the right of x = c, then the function is changing from concave up to concave down. This is an inflection point.

If f”(x) < 0 on the left of x = c, and f”(x) > 0 on the right of x = c, then the function is changing from concave down to concave up. This is an inflection point.

If $f”(x)$ is the same sign on either side of $x = c$, then the function does not change concavity at $c$, and $c$ is not an inflection point.

Examples of Critical Points

Let’s walk through some critical point examples.

Critical Points of a Function

Identify all critical points of the polynomial $f(x) = x^3 + 2x^2$.

Step 1

The first step is to take the first derivative of the given function. Using the power rule for derivatives, $f’(x) = 3x^2 + 4x$.

Step 2

The second step is to set the first derivative equal to zero and solve. By factoring, this gives us our critical numbers. Note there are no points where the first derivative is undefined in this case.

$3x^2 + 4x = 0$

$x(3x + 4) = 0$

$x = 0, x = -\frac{4}{3}$

Step 3

The third step is to plug our critical numbers into our original function. This gives us the y-coordinates of our critical points. So, our critical points are $(0, 0)$ and $(-\frac{4}{3}, \frac{32}{27})$.

Critical Points on a Graph

We can also identify critical points on the graph of a function. Consider the below graph of the function $y = \frac{x^4}{2} + 2x^3 + 4$. Can you identify its critical points?

A local minimum is at $x = -3$, and so we have a critical point at $(-3, -9.5)$.

While there are no further relative extrema, we can make another horizontal tangent line at $x = 0$ where the function changes concavity. So, we have another critical point at $(0, 4)$. This is an inflection point.

Note that there is another inflection point at $x = -2$. However, this is not a critical number since there is no horizontal or vertical tangent here.

Critical Points of Multivariable Functions

In multivariable calculus, functions have more than one variable. The critical points of the function $z = f(x, y)$ are the points $(a, b)$ such that the partial derivatives$f_x(a, b) = 0$ and $f_y(a, b) = 0$. Just like one-variable functions, finding a critical point of a multivariable function is not a guarantee relative extrema is at that point.

As with single-variable calculus, there are 3 kinds of critical values:

Relative Maxima: If $f(a, b) >= f(x, y)$ for all points $(x, y)$ in the region, then $f(a, b)$ is a relative maximum.

Relative Minima: If f(a, b) <= f(x, y) for all points (x, y) in the region, then f(a, b) is a relative minimum.

Saddle Point: Saddle points are not local extrema. They appear as a relative maximum when approached from one direction, and a relative minimum when approached from the other direction. Below is the saddle point of $z = x^2 - y^2$.

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