Critical Points in Single Variable Calculus
In this section, we will assume that we're working with differentiable functions only.
In single-variable calculus, a critical point is a point where a function's first derivative is zero or undefined.
First derivative as slope
To understand critical points, we need have a sense of what a derivative is.
The derivative of a function describes the way one quantity changes in relationship to another. Let's say we have a function that gives the population size of a community of rabbits based on the number of wolves living in the region. The derivative of this function would tell us how the rabbit population changes as the wolf population changes. If the derivative is negative for some number of wolves, that means that as more wolves enter the area, the rabbit population goes down. If the derivative is positive, it means that as more wolves enter the area, the rabbit population goes up. If the derivative is zero, as it is at a critical point, that wolf population produces a stable rabbit population - an equilibrium.
When we graph functions, we usually think of the first derivative as being the slope of the tangent line. A tangent line is a line that shares only one point in common with the function you're looking at. The tangent line measures the way the y value changes with respect to the x value. The image below is a graph of the parabola , with tangent lines drawn in at the points , , and . If we connect this image to our population example, the x values would be the size of the wolf population, the y values would be the size of the rabbit population, and the derivatives give us the rates of change.
At , the first derivative is , and so is the slope of the tangent line. At , the derivative and the slope are both . At , the line is horizontal, demonstrating that the derivative at this point is . Since the tangent line there has a slope of , the point is a critical point of the function .
A good way to find critical points on a graph, therefore, is to find points where a tangent line would have a slope of , or where a tangent line would be vertical (this corresponds to the derivative being undefined).
First derivative test
Critical points are closely related to the first derivative test, which uses critical points and slopes to find local maximums and minimums of a function without graphing the function.
Critical Points in Multivariable Calculus
In multivariable calculus, a critical point is a point where a function's gradient is either 0 or undefined.
Finding critical points
To find the critical points of a function in several variables, we first find the function's gradient. Points for which the gradient is zero or undefined are critical points. If the partial derivatives are defined everywhere, we can simply set each partial derivative in the gradient equal to zero, and solve the system of equations. If the partials aren't defined everywhere, we need to identify every point where either one is undefined.
Let's try this with the function below:
First, we find the gradient:
Since both partials are polynomials, they're defined everywhere, so we set them equal to zero and solve:
So our function has a critical point where and . To identify the actual point, we plug these values back into and solve for the coordinate of that point:
The critical point, then, is at . This point is shown in the graph below:
Classifying critical points
As in single variable calculus, multivariable critical points are typically local maximums or minimums of the function.