Thread: Find extremal points of f(x,y)

1. 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):

$f(x,y)=x^2 + y^2 + 2x + 4y - 8 \sqrt{x y} + 4$

2. Re: Find extremal points of f(x,y)

What did you try?

Starty by solving $f_x=0$ and $f_y=0$.

3. 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.

4. 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) ?

5. Re: Find extremal points of f(x,y)

I think not. Can you solve it?

6. 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 ?