Solving an inequality with AM-GM

Question

$X+Y>N$ and $$X+Y+2\sqrt{(X+13)(Y+26)}>-14$$ Find $N$.

Got really confused... Should I use AM-GM which $$2\sqrt{(X+13)(Y+26)}<(X+13)+(Y+26)$$ (since we want to minimize $M+N$) we maximize $$2\sqrt{(X+13)(Y+26)}$$? Got really confused. Please help!

Answer 1

Since $2(a^2+b^2)\geq(a+b)^2$, we obtain: $$2(x+13+y+26)\geq\left(\sqrt{x+13}+\sqrt{y+26}\right)^2>25.$$ Thus, $x+y>-26.5$.

For $x\rightarrow-\frac{27}{4}$ and $y\rightarrow-\frac{79}{4}$ say that our estimation was exact.

Thus, a best $N=-26.5$.