Question

how to find the exterma of the function F(x,y)=sinxy

Answer 1

Differentiate twice: one time for x and one for y.
Then have both these = 0. Solve this system for extrema.

Answer 2

I mean how to find the absolute exterma for this function...?

Answer 3

I don't know if I use correct math-grammar, but tell me if you don't get me. (Not an English-math-learner)

Answer 4

No neoag it was great thanks:)

Answer 5

Well. if you do it my way, you will find all of them. Then you probably can see which one is "the lowest value" and highest. Also, for a function to have the property of having an absolute extrema - you need specific criteria. You need to have a compact interval to guarantee such a value.

Answer 6

F'x=y*cos(xy) will equal 0 for y=0 AND xy=pi/2 + n*pi
F'y=x*cos(xy) will equal 0 for x=0 AND xy=pi/2 + n*pi

So you have extrema whenever xy=pi/2+n*pi and in (0,0).

Now, if you look at the F-function when these conditions are fulfilled.
We have F=sin(0) and F=sin(pi/2+n*pi/2).

This means that F has a maximal and minimal value of 1 and -1. I think.

Answer 7

That seems right.. I only forgot to add the closed region the question had gave

0<=x=<pi

0<=y=<1

Answer 8

In my last post where I wrote F=sin(pi/2+n*pi/2), it should be F=sin(pi/2+n*pi).
You should control where the extrema INSIDE this area, and on the borders.
Check the borders by checking x=0 and the range 0<=y=<1, then x=pi with 0<=y=<1
similar with y. Check for extrema, (you can reduce F to one variable and derivate it normally when checking borders.)

Then control the inside in the same way as I did.

Answer 9

thanks so much neogh:)