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PD.7 Maximum and Minimum Values

 

As you have seen before, a main use of the derivative is to find a maximum and minimum values. In this section we will see how to use partial derivatives to locate maxima and minima of functions of two variables. First we will start out by formally defining local maximums and minimums:

 

A function of two variables has a local maximum at $(a,b)$ if $f(x,y)\leq f(a,b)$ when $(x,y)$ is near $(a,b).$ [This means that $f(x,y)\leq f(a,b)$ for all points $(x,y)$ in some disk with center $(a,b).]$ The number $f(a,b)$ is called a local maximum value.. If $f(x,y)\geq f(a,b)$ when $(x,y)$ is near $(a,b),$then $f(a,b)$ is a local minimum value. If the inequalities hold true for all points $(x,y)$ in the domain of $f$, then $f$ has an absolute maximum (or absolute minimum ) at $(a,b)$.

 

Example: The graph shows the critical points of the function. The animations will allow you to see all the critical points of the graph.


 


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The process for finding these maxima and minima is similar to the one variable process, just set the derivative equal to zero. However, using two variables, one needs to use a system of equations. This process is given below in the following theorem:

If $f$ has a local maximum or minimum at $(a,b)$ and the first-order partial derivatives of $f$ exist there, then $f_{x}(a,b)=0$ and $f_{y}(a,b)=0.$ These points, where $f_{x}(a,b)=0$ and $f_{y}(a,b)=0$ are called critical points.

 



 

We treat these critical points just like we do in single variable calculus, they can be an absolute maximum value, a local maximum value, an absolute minimum value, a local minimum value, or they can be neither. However, the point can also be called a saddle point. Regardless, the one thing that all of these points have in common is that they all have a slope of zero.

Just as in single variable calculus, we can use the second derivative to help us determine what the critical points will be classified as. However, the second derivative test in two variables is more complicated that when using only one variable:

Suppose the second partial derivatives of $f$ are continuous on a disk with center $(a,b)$ and suppose that $f_{x}(a,b)=0$ and $f_{y}(a,b)=0$ (in other words $(a,b)$ is a critical point) then:


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Then:
(a) If $D>0$ and $f_{xx}(a,b)>0$, then $f(a,b)$ is a local minimum
(b) If $D>0$ and $f_{xx}(a,b)<0$, then $f(a,b)$ is a local maximum
(c) If $D<0$ then $f(a,b)$ is not a local maximum or minimum.
 

 

Notes

1. In case (c) the point $(a,b)$ is called a saddle point of $f$ (see picture below).

2. If $D=0$, the test fives no information: $f$ could be anything.

3. To remember the formula, it is helpful to write is as a determinant:


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Absolute Maximum and Minimum Values

To find absolute maximum and minimum values in two variables, we need to define the area that is finite. This area is called a bounded set. A bounded set in $\U{211d} ^{2}$ is one that is contained with in some disk$.$In other words, it is finite in extent.

 

Extreme Value theorem for functions of two variables.


If $f$ is continuous on a closed, bounded set $D$ in $\U{211d} ^{2},$ then $f$ attains an absolute maximum value $f(x_{1},y_{1})$ and an absolute minimum  value $f(x_{2},y_{2})$ at some points $(x_{1},y_{1})$ and $(x_{2},y_{2}).$
 

 

To find the extreme values guaranteed by the above theorem, we note that, if $f$ has an extreme value at $(x_{1},y_{1})$, then $(x_{1},y_{1})$ is either a critical point of $f$ or a boundary point of $D$ Thus, we have the following extension of the theorem:

 

To find the absolute maximum and minimum values of a continuous function $f$ on a closed, bounded set $D$:
1. Find the values of $f$ at the critical points of $f$ in $D$.
2. Find the extreme values of $f$ on the boundary of $D$.
3. The largest of the values from steps 1 and 2 is the absolute maximum value; the smallest of these values is the absolute minimum value.
 

Example: The function has absolute maximum and absolute minimum. Use the animation to rotate the graph and discover  the extrema.

 


 

  


Maximum and Minimum Values and Contour Map

 

Example: In this animation the function is graph with the contour map. Observe the relation between the critical points and the contour curves.

 

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