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Math Help - Local Max/Min Values using Partial Derivatives

  1. #1
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    Local Max/Min Values using Partial Derivatives

    f(x, y) = 2x^3 + xy^2 + 5x^2 + y^2 + 8

    Find the local max, min and saddle points of the function.

    So far, I have:
    -partial derivative of x= 6x^2 + y^2 + 10x
    -partial derivative of y= 2xy+2y

    But I'm really confused what to do next. My book says to set both of the partial derivative equations to 0 and then solve for one of the variables, but I get some weird answer that doesn't work in both equations.

    Thanks so much for your help!
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  2. #2
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    Quote Originally Posted by juicysharpie View Post
    f(x, y) = 2x^3 + xy^2 + 5x^2 + y^2 + 8

    Find the local max, min and saddle points of the function.

    So far, I have:
    -partial derivative of x= 6x^2 + y^2 + 10x
    -partial derivative of y= 2xy+2y

    But I'm really confused what to do next. My book says to set both of the partial derivative equations to 0 and then solve for one of the variables, but I get some weird answer that doesn't work in both equations.

    Thanks so much for your help!

    So you need to solve this system of equaitons

    6x^2+y^2+10x=0 and

    2xy+2y=0

    Notice that we can factor the bottom equaiton to get

    2y(x+1)=0 so by the zero factor principle we get

    either 2y=0 or x+1=0

    So now if y=0 we can plug that into the top equation to get 6x^2+10x=0 \iff 2x(3x+5)=0 si we get two possible ordered pairs (0,0) \mbox{ or } (-\frac{5}{3},0)

    You need to use the other solution for x above to find the rest of your critical points. I hope this helps
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  3. #3
    MHF Contributor arbolis's Avatar
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    The first step is as the book says and as TheEmptySet started you on. Once you have all the critical points you're not done! You have to show whether they are local max/min or saddle point.
    Call D=f_{xx}(a,b)f_{yy}(a,b)-f_{xxy}(a,b).
    Now, if D>0 and f_{xx}(a,b)>0, then (a,b) is a local minimum.
    If D>0 and f_{xx}(a,b)<0 then (a,b) is a local maximum.
    If D<0 then (a,b) is a saddle point.

    Finally, if D>0 and f_{xx}(a,b)=0 then you cannot conclude using this formula, and I don't really know how to proceed. I hope your exercise doesn't fall into this category.
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