How can I prove that $xy\leq x^2+y^2$?

$$x^2+y^2-xy=\frac{x^2}{2}+\frac{y^2}{2}+\frac{(x-y)^2}{2}$$


Use polar coordinates: $$x=r \cos \theta,y=r \sin \theta $$

Your inequality becomes $$r^2 \cos \theta \sin \theta \leq r^2 $$

which is pretty much trivial.


Though there are quite a few proofs already given, I'd like to add a visual one.

picture

Tags:

Inequality

Real Analysis

Related

Calculating probabilities over different time intervals If $a_1,a_2,\dotsc,a_n>0 $, then $\lim\limits_{x \to \infty} \left[\frac {a_1^{1/x}+a_2^{1/x}+\dotsb+a_n^{1/x}}{n}\right]^{nx}=a_1 a_2 \dotsb a_n$ $f^{-1}(U)$ is regular open set in $X$ for regular open set $U$ in $Y$, whenever $f$ is continuous. Proving that $\sum\limits_{k=0}^{n} {{m+k} \choose{m}} = { m+n+1 \choose m+1 }$ Is $2^{218!} +1$ prime? What's the point of a Horseshoe map? Is there a short proof for the Intermediate Value Theorem Showing an indentity with a cyclic sum Counting non-isomorphic relations Orientability of projective space Generalizing Ramanujan's proof of Bertrand's Postulate: Can Ramanujan's approach be used to show a prime between $4x$ and $5x$ for $x \ge 3$ How do i prove that $\det(tI-A)$ is a polynomial?

Recent Posts