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Find Absolute Max/Min for Function with Domain.

  1. Nov 1, 2009 #1
    1. Find absolute max/min for the given function with the specified domain.
    f(x,y)= x^2-xy+y^2 on the circular disk x^2+y^2 less/equal 4, use parametrization when checking boundary.



    2. f(x,y)= x^2-xy+y^2 x^2+y^2 less/equal 4



    3. The only thing I understand so far is that I must change the boundary to parametric equation. (x= rcos(t) and y=rsin(t)). I think t is bounded openly between 0 and 2. Also I think i use the chain rule to find equation and set it in terms of t to check points. Help Please!
     
  2. jcsd
  3. Nov 1, 2009 #2

    jambaugh

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    Visualize the function as defining height above the x,y plane what does a local maximum or local minimum look like (thing in terms of geographic features). If you are walking in uneven territory and are blind how do you know if you are standing on a local maximum or local minimum rather than somewhere else?

    An absolute maximum or minimum must occur either at a local maximum/minimum or on the boundary. How do you find local maxima and minima? How did you do it in single variable calculus?

    Hint, why is this question being asked in a calculus class.

    As far as your boundary is concerned remember that from its parametric form you can derive the quantity to be optimized q=f(x,y) as a function of the parameter q = h(t) = f( x(t), y(t) ).
    You then again have another optimization problem what is the max and min of q on the boundary?
     
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