1
$\begingroup$

How can I find the maxima and minima of the function $f:\mathbb{RP}^2 \rightarrow \mathbb{R},\space f([x,y,z])=\frac{xy+yz+2zx}{x^2+y^2+z^2}$ ?

I believe that I have to use the Lagrange multipliers to get the desired result, but I have absolutely no idea how.

  • 0
    Think about when the function reaches a maximum and a minimum for example g(x,y,z) = 0, where g = xy+yz+2zx. What are the constraints? If you can identify them can pose it as a linear programming problem it is essentially using lagrange mulitpliers – 2017-01-19
  • 0
    The weird thing is that the exercise was formulated exactly as above, without any constraint! I didn't catch the meaning of your first sentence, can you repeat it otherwise? – 2017-01-19
  • 0
    The function f reaches a maximum when the denominator reaches zero, so essentially $\lim\limits_{something\rightarrow 0} (x^2+y^2+z^2)$ is a constraint for maximum. Hope this helps – 2017-01-19
  • 0
    how would I use this when the notion of limits on manifolds is non-existent? – 2017-01-19

1 Answers 1

2

The function is homogeneous of degree $0$ - in other words, scaling all the variables by $\lambda$ does not change the value of the function. So, you can normalize by setting $x^2 + y^2 + z^2=1.$ Now, Lagrange multipliers are your friend.

  • 0
    Thank you, that helped me a lot, actually! But, computing the gradient of $f$ won't be a problem since the function takes values from $\mathbb{RP}^2$? – 2017-01-19
  • 0
    Oh wait, we don't have to care about the domain of the function, since I want to maximize the numeric expression $xy+yz+2zx$ when $x^2+y^2+z^2=1$ and all of these are...well, numbers. So I can work with functions that have as domain the $\mathbb{R}^3$! Am I right? – 2017-01-19
  • 0
    @SotirisSimos Correct. – 2017-01-19