zmudz one year ago For x, y, and z positive real numbers, what is the maximum possible value for $$\sqrt{\frac{3x+4y}{6x+5y+4z}} + \sqrt{\frac{y+2z}{6x+5y+4z}} + \sqrt{\frac{2z+3x}{6x+5y+4z}}?$$ Also, find what z/x is if (x,y,z) achieves the maximum value. Thank you! I am so stuck right now.

1. ganeshie8

Let $$a=\sqrt{\frac{3x+4y}{6x+5y+4z}}\\ b=\sqrt{\frac{y+2z}{6x+5y+4z}}\\ c=\sqrt{\frac{2z+3x}{6x+5y+4z}}$$ Firstly, notice that $$a^2+b^2+c^2=1$$

2. ganeshie8

appealing to cauchy schwarz inequality gives $a*1+b*1+c*1\le \sqrt{(a^2+b^2+c^2)(1^2+1^2+1^2)}=\sqrt{3}$