x\sqrt{1-x^{2}}+y\sqrt{1-y^{2}}=\sqrt{x}\times \sqrt{x-x^{3}}+\sqrt{y}\times \sqrt{y-y^{3}}\leq \sqrt{1-(x^{3}+y^{3})} (BĐT AM-GM) Mà 1-(x^{3}+y^{3})=(x+y)^{3}-(x^{3}+y^{3})=3xy(x+y)=3xy\leq3\times (\frac{x+y)}{2})^{2}=\frac{3}{4}\Rightarrow đpcm
$x\sqrt{1-x^{2}}+y\sqrt{1-y^{2}}=\sqrt{x}\times \sqrt{x-x^{3}}+\sqrt{y}\times \sqrt{y-y^{3}}\leq \sqrt{1-(x^{3}+y^{3})} (BĐT AM-GM)
$ Mà
$1-(x^{3}+y^{3})=(x+y)^{3}-(x^{3}+y^{3})=3xy(x+y)=3xy\leq3\times (\frac{x+y)}{2})^{2}=\frac{3}{4}
$$\Rightarrow
$ đpcm