Thread: 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):

What did you try?

Starty by solving and .

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



At least I don't see I way how to get rid of those pesky square roots.

From fx=0, we have ... (1)

which gives ... (2)

From fy=0, we have ... (3)

Putting (1) & (2) in (3) gives :

which can be written as ... (4)

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

as

Let t=x+1, or :

... (5)

Can you solve (5) ?

I think not. Can you solve it?

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 ?