Processing math: 100%

Sunday, November 27, 2022

2022/073) Maximize x \sqrt{1-y^2} + \sqrt{1-x^2}

 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}

 


 

No comments: