show that among all positive numbers x and y with x^2 + y^2 = r^2, the sum of x+y is largest when x=y=r/(sqrt 2)

S=x+y=x+v(r²-x²)

Ds/dx=1+(-2x)/2v(r²-x²) put ds/sx=0 , we get

1-x/v(r²-x²)=0

ð X=r/v2

ð D²s/dx²=0- { v(r²-x²).1+x²/v(r²-x²)}

ð D²/dx²=-r²/v(r²-x²)

Pur x=r/v2, we get

Y=v(r²-x²)

=v(r²-r²/2

=vr²/2

=r/v2

So x=y=r/v2

Thanks & regards

Rinkoo Gupta

askIITians faculty