Critical Points

Learning Objectives

  • Use partial derivatives to locate critical points for a function of two variables.

For functions of a single variable, we defined critical points as the values of the function when the derivative equals zero or does not exist. For functions of two or more variables, the concept is essentially the same, except for the fact that we are now working with partial derivatives.

Definition


Let [latex]z=f(x, y)[/latex] be a function of two variables that is defined on an open set containing the point [latex](x_0, y_0)[/latex]. The point [latex](x_0, y_0)[/latex] is called a critical point of a function of two variables [latex]f[/latex] if one of the two following conditions holds:

  1. [latex]f_x=(x_0, y_0)=f_y(x_0, y_0)=0[/latex]
  2. Either [latex]f_x(x_0, y_0)[/latex] or [latex]f_y(x_0, y_0)[/latex] does not exist.

Example: Finding critical points

Find the critical points of each of the following functions:

a. [latex]f(x,y)=\sqrt{4y^2-9x^2+24y+36x+36}[/latex]

b. [latex]g(x,y)=x^2+2xy-4y^2+4x-6y+4[/latex]

try it

Find the critical point of the function [latex]f(x, y)=x^{3}+2xy-2x-4y[/latex].

Watch the following video to see the worked solution to the above Try It

The main purpose for determining critical points is to locate relative maxima and minima, as in single-variable calculus. When working with a function of one variable, the definition of a local extremum involves finding an interval around the critical point such that the function value is either greater than or less than all the other function values in that interval. When working with a function of two or more variables, we work with an open disk around the point.

Definition


Let [latex]z=f(x, y)[/latex] be a function of two variables that is defined and continuous on an open set containing the point [latex](x_0, y_0)[/latex]. Then [latex]f[/latex] has a local maximum at [latex](x_0, y_0)[/latex] if

[latex]f(x_0, y_0)\geq f(x, y)[/latex]

for all points [latex](x, y)[/latex] within some disk centered at [latex](x_0, y_0)[/latex]. The number [latex]f(x_0, y_0)[/latex] is called a local maximum value. If the preceding inequality holds for every point [latex](x, y)[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] has a global maximum (also called an absolute maximum) at [latex](x_0, y_0)[/latex].

The function [latex]f[/latex] has a local minimum at [latex](x_0, y_0)[/latex] if

[latex]f(x_0, y_0)\leq f(x, y)[/latex]

for all points [latex](x, y)[/latex] within some disk centered at [latex](x_0, y_0)[/latex]. The number [latex]f(x_0, y_0)[/latex] is called a local minimum value. If the preceding inequality holds for every point [latex](x, y)[/latex] in the domain of [latex]f[/latex], then [latex]f[/latex] has a global minimum (also called an absolute minimum) at [latex](x_0, y_0)[/latex].

If [latex]f(x_0, y_0)[/latex] is either a local maximum or local minimum value, then it is called a local extremum (see the following figure).

The function z = the square root of (16 – x2 – y2) is shown, which is the upper hemisphere of radius 4 with center at the origin. In the xy plane, the circle with radius 4 and center at the origin is highlighted; it has equation x2 + y2 = 16.

Figure 2. The graph of [latex]\small{z=\sqrt{16-x^{2}-y^{2}}}[/latex] has a maximum value when [latex]\small{(x,y)=(0,0)}[/latex] It attains its minimum value at the boundary of its domain, which is the circle [latex]\small{x^{2}+y^{2}=16}[/latex].

In Maxima and Minima, we showed that extrema of functions of one variable occur at critical points. The same is true for functions of more than one variable, as stated in the following theorem.

Theorem: Fermat’s theorem for functions of two variables


Let [latex]z=f(x, y)[/latex] be a function of two variables that is defined and continuous on an open set containing the point [latex](x_0, y_0)[/latex]. Suppose [latex]f_x[/latex] and [latex]f_y[/latex] each exists at [latex](x_0, y_0)[/latex]. If [latex]f[/latex] has a local extremum at [latex](x_0, y_0)[/latex], then [latex](x_0, y_0)[/latex] is a critical point of [latex]f[/latex].