<s>[INST] Find Minimum Value of: $P=x^2+y^2+2xy+2-\frac{1}{xy}$ Given: $x,y>0$ and $x^2y-x+xy^2-y-3xy=0$
Find Minimum Value of:
$P=x^2+y^2+2xy+2-\frac{1}{xy}$
I found $\min P =\frac{71}{4}$ at $x=y=2$ but I cant prove that
Could some one help me ?
 [/INST] It is sure that, without using Lagrange multipliers, the problem is quite hard.
But let us try using brute force. The constraint $x^2y-x+xy^2-y-3xy=0$ gives a quadratic equation in $y$; so we can express $y$ as a function of $x$ $$y_{\pm}=\frac{-x^2+3 x+1\pm\sqrt{x^4-6 x^3+11 x^2+6 x+1}}{2 x}$$ Replace $y_+$ in $P$ (which is now an ugly function of $x$) and plot it (even better, compute $P'_x$ and plot it to find the root). You will effectively find $x=2$ to which corresponds $y=2$ and $P=\frac{71}{4}$.
Doing the same with $y_-$ shows a minimum for $x\approx 1.54426$ corresponding to $y \approx-0.399545$ which has to be discarded.
</s>