Extrema of functions of Two Variables

[CalleighMay]

Hello, my name is Calleigh and i am new to the forum! I am in Calculus II and have a few questions on some problems. I am using the textbook Calculus 8th edition by Larson, Hostetler and Edwards. Could someone please help me?

 

The problem is on pg 959 in chapter 13.8 in the text, number 56. It reads:

 

a. Find the absolute extrema of the function.

b. From the form of he function, determine whether a relative maximum of a relative minimum occurs at each point.

 

and it gives:

f(x,y)=2x-2xy+y^2

and R:The region in the xy-plane bounded by the graphs of y=x^2 and y=1

 

What does it mean when it states f(x,y)??? I've already heard of f(x). I don't know what it means when it has both x and y in the parenthesis. My professor gave these problems to us even though we haven't covered it yet, but he expects us to know how to do it?? Please help me, i'm so lost...

 

 

Any help would be greatly appreciated! Thanks guyssss

[Bignose]

A function of two variables is just like a function of one variable, it just has two inputs instead of one.

 

It may be easier to see this if you fix several values of one of that variables. "Lock" in a value of y at Y1, and look at f(x,Y1). This looks just like an f(x) value once you plug in Y1. Now, plug in a different value Y2. And another Y3. The function f(x) (which will really be f(x,Y2) or f(x,Y3)) will change, but still look just like a single variable function.

 

It is just the natural extension of the single variable function. Because nature and life and physics have things that depend on more than one variable. The velocity of air flowing along the ground is going to depend on many variables -- at the very least it is going to depend on the longitude and latitude of where you measure it. And, it is going to depend on how high above the ground you measure it (altitude) as well. So the wind velocity, v, is a function of at least three variables. v(long., lat., alt.) Now, if you wanted to model it, there could even be more variables. Like the temperature and pressure. And the gradients thereof. I.e. if a high pressure front is moving into a low pressure front, there is a steep gradient of pressure, so the wind is going to be higher than if there is a gentle gradient of pressure. So, wind velocity so far could be a funciton of at least 7 variables, now: v(long., lat., alt., temp., temp. gradient, press., press. gradient)

 

And there are potentially many others (like humidity) but I think you see the point.

 

Let me give you a very different example. What variables would go into writing a function of the average income of a person? There is pretty much no way to capture this with only one variable because of all the competing influences. You'd have to include level of education (a college grad on average makes more than a high school dropout), you'd have to include geographic location (the cost of living in a coastal state is higher than a Midwestern state), you'd have to include gender, race, maybe religion. You may consider including a variable that includes one's parents (i.e. a child whose parents both went to college usually ends up with a better job that a child whose parents both dropped out of high school). And there are probably many more. But, again, the point is that the result -- the average income of a person -- is going to be a function of many, many different things. These cannot be captured by only one variable f(x). Usually, you are going to need f(x,y,z,a,b,c,...).

 

The big thing is that it actually is a pretty special case where only one variable drives the resulting output. Of course, you study the one variable cases extensively to start with to learn the basics and keep confusion to a minimum. But, the f(x,y,z,...) is just the natural extension of f(x) to acknowledge that very often in real life more than one variable influences the final outcomes.

[booker]

What does it mean when it states f(x,y)??? I've already heard of f(x). I don't know what it means when it has both x and y in the parenthesis. My professor gave these problems to us even though we haven't covered it yet, but he expects us to know how to do it?? Please help me, i'm so lost...

 

Be sure someone tells him so.

 

You can think of f(x,y) as the height of a surface about the x-y plane.

 

Partial derivatives are geometrically simple to envision. (If you haven't had partial derivative yet, your professor needs some additional reminding.) Slice through the surface along at some particular constant x.

 

[math]\partial f / \partial x [/math] will be the slope of the surface along the cut. the value of [math]\partial f / \partial x [/math] is dependent on the value of y where the cut was taken.

 

The same argument applies to [math]\partial f / \partial y [/math].

Edited August 3, 2008 by booker

[CalleighMay]

Thanks for the replies guys ;P I understand now what it means when there are more than one variable in the parenthesis, Bignose your analogy was perfect

 

Our professor just gave us these problems so we could "preview" what we would be seeing next semester. He just wants to work together to see what we come up with, all of my friends have no idea, i just really want to impress my professor!

 

So Booker you're saying i should begin by taking the derivative of the function? I took the deriv using my calculator and got 2, something tells me this is wrong since there's probably a y' in there somewhere? =/

 

well tonight my friends and i looked ahead and tried to learn the stuff on our own, and we gave it a try by looking at some examples in the book and this is what we came up with.

 

We got:

 

f(x,y)=2x-2xy+y^2

derivative with respect to x is f sub x =6x=0, x=0

with respect to y, 4y-4=0, y=1

then f(0,1)=-2

on line line of y=4, -2<=x<=2

so, f(x,y)=f(x)=3x^2+32-16=3x^2-16

and the max would be at 28 with a minimum at 16

on the curve of y=x^2, -2<=x<=2

f(x,y)=f(x)=3x^2+2(x^2)^2-4x^2=2x^4-x^2=x^2[2(x^2)-1]

and the max would be at 28 and the minimum at -1/8

 

absolute max 28 at (+ or -2, 4)

absolute min -2 at (0,1)

 

Does this seem about right? thanks