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Math Help - Find max/min values.

  1. #1
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    Find max/min values.

    Find the max and min values of f(x,y)=x^2 + y^2 + 4 over the region R={(x,y) : x^2 + 2(y^2) =< 3
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  2. #2
    MHF Contributor MarkFL's Avatar
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    Re: Find max/min values.

    You want to equate the first partials to zero and solve the resulting system to find the critical points:

    f_x(x,y)=0

    f_y(x,y)=0

    Solve the resulting system to find any critical points in the given region.

    Now you want to use the second partials test for relative extrema:

    Let (a,b) be a critical point of z=f(x,y) and suppose f_{xx},\,f_{yy} and f_{xy} are continuous in a rectangular region containing (a,b).

    Let D(x,y)=f_{xx}(x,y)f_{yy}(x,y)-(f_{xy}(x,y))^2.

    i) If D(a,b)>0 and f_{xx}(a,b)>0, then f(a,b) is a relative minimum.

    ii) If D(a,b)>0 and f_{xx}(a,b)<0, then f(a,b) is a relative maximum.

    iii) If D(a,b)<0, then f(a,b) is not an extremum.

    iv) If D(a,b)=0, then no conclusion can be drawn concerning a relative extremum.
    Last edited by MarkFL; November 15th 2012 at 08:01 PM.
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