Proving an inequality of two variables

Question

The inequality is the following : $\frac{x^2+y^2}{4} \leq e^{x+y-2}$, where $x,y \geq0$.

I tried manipulating the inequality using Taylor series, but I couldn't find a conclusive result . Any ideas ?

Answer 1

Hint: $x=\rho \cos \theta$, $y=\rho \sin \theta$, with $\theta \in [0;\dfrac{\pi}{2}]$ and $\rho \geq 0$

Then $\dfrac{x^2+y^2}{4}=\dfrac{\rho^2}{4}$ and $e^{x+y-2}=\dfrac{e^{\rho (\cos \theta + \sin \theta)}}{e^2}$

Can you conclude something by comparing the two expression written as such?