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Math Help - Find extremal points of f(x,y)

  1. #1
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    Find extremal points of f(x,y)

    I've solved hundreds of these kind of problems and then I came across this one. Well, normally, I'd say that it cannot be solved in closed form, only to realise that it is given to first year undergraduate students. So I must be overseeing something

    Find extremal points of f(x,y):

    f(x,y)=x^2 + y^2 + 2x + 4y - 8 \sqrt{x y} + 4
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  2. #2
    Ted
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    Re: Find extremal points of f(x,y)

    What did you try?

    Starty by solving f_x=0 and f_y=0.
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  3. #3
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    Re: Find extremal points of f(x,y)

    Well, that's the whole point. It can't be solved in closed form:

    2x + 2 - 4 \sqrt{y/x} = 2y + 4 - 4 \sqrt{x/y}=0

    At least I don't see I way how to get rid of those pesky square roots.
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  4. #4
    Ted
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    Re: Find extremal points of f(x,y)

    From fx=0, we have \displaystyle \sqrt{\frac{x}{y}}=\frac{2}{x+1} ... (1)

    which gives \displaystyle y=\frac{1}{4} x (x+1)^2 ... (2)

    From fy=0, we have \displaystyle 2 \sqrt{\frac{x}{y}} = y +2 ... (3)

    Putting (1) & (2) in (3) gives :  \frac{4}{x+1} = \frac{x}{4} (x+1)^2 + 2

    which can be written as  16 = x(x+1)^3+8(x+1) ... (4)

    You can expand (4) and solve for x, but to make our life easier, (4) can be written

    as 16=(x+1-1)(x+1)^3+8(x+1)

    Let t=x+1, 16=(t-1)t^3+8t or :

    t^4-t^3+8t-16=0 ... (5)

    Can you solve (5) ?
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  5. #5
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    Re: Find extremal points of f(x,y)

    I think not. Can you solve it?
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  6. #6
    Ted
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    Re: Find extremal points of f(x,y)

    I'd start by guessuing a number t=a which satisfies (5).
    There is a well-known method for finding such number, provided that (5) has real solution(s).
    Do you know it ?
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