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Let's have: $$f(x,y)=\max\{x^2+2y,y^2+4x\}\forall x,y\in\mathbb R$$ We need to find the least value of $f(x+y)$, Now let us choose $*$ to denote any of $\ge,\le,=,<,>$. Here: $$x^2+2y*y^2+4x\implies \frac{(x-2)^2}{\sqrt3^2}-\frac{(y-1)^2}{\sqrt3^2}*0$$ which denotes regions of a hyperbola with centre at $2,1$. I do not get what can we do next.

shinjin
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RE60K
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  • I have seen multivariable maxima and minima, usually have an extra constraint set up usually a path associated. For e.g, http://math.stackexchange.com/questions/1277607/global-maximum-and-minimum-of-fx-y-z-xyz-with-the-constraint-x22y23z2#comment2594511_1277607 , probably try take the gradient and see if anything helps :/ – Someone May 12 '15 at 14:45
  • @Mann no Lagarnage multipliers. – RE60K May 12 '15 at 14:47

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