<s>[INST] Find minimum value of $P=17x^2+17y^2+16xy$ Let $x,y$ be positive real numbers such that $4x^2+4y^2+17xy+5x+5y\ge 1$. Find minimum value of $$P=17x^2+17y^2+16xy$$

My idea:
I see that $x=y=\frac{\sqrt 2 -1}{5}\rightarrow P=2(3-2\sqrt 2)$
So I proved $P\ge 2(3-2 \sqrt 2)(4x^2+4y^2+17xy+5x+5y)\ge2(3-2\sqrt 2)$
Is my approachment is true?
 [/INST] Yes, your answer is true.
The solution can be the following.
By AM-GM
$$1\leq4(x+y)^2+5(x+y)+9xy\leq4(x+y)^2+5(x+y)+\frac{9}{4}(x+y)^2,$$ which gives
$$x+y\geq\frac{2(\sqrt2-1)}{5}.$$
Thus, by C-S we obtain:
$$17(x^2+y^2)+16xy=8(x+y)^2+\frac{9}{2}(1^2+1^2)(x^2+y^2)\geq8(x+y)^2+\frac{9}{2}(x+y)^2=$$
$$=\frac{25}{2}(x+y)^2\geq\frac{25}{2}\cdot\frac{4(3-2\sqrt2)}{25}=2(3-2\sqrt2).$$
The equality occurs for $x=y=\frac{\sqrt2-1}{5},$ which says that we got a minimal value.
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