In the above expression x not dependent on y term so let us maximize \sqrt{1-y^2} and this is {maxium at y= 0 and the value is 1.
Putting the same we get x + \sqrt{1-x^2}
As we have \sqrt{1-x^2} we can put x=\sin\,t ( 0 \le t \le \frac{\pi}{4} to get
x + \sqrt{1-x^2}= \sin\, t + \cos \, t
We need to maximize \sin\, t + \cos \, t so let us get in form r \cos\, w \sin \, t + r \sin\, w \cos \, t
which is r \sin ( t+w) and maximum value is |sqrt{2}$
So let r\cos\, t = 1 and r \sin\, t = 1
By squaring both and adding
r^2 = 2 or r=\sqrt{2}
Hence largest valus is \sqrt{2}
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