# Math Help - How can i show that ...

1. ## How can i show that ...

If $a + b + c + d = 1 \Rightarrow$

Show that $\sqrt{(4a + 1)} + \sqrt{(4b + 1)} + \sqrt{(4c + 1)} + \sqrt{(4d + 1)} \le 6$

2. Hello,

Maybe I have an idea :

$a+b+c+d=1\Leftrightarrow (4a+1)+(4b+1)+(4c+1)+(4d+1)=8$

And if we write $\left\{\begin{array}{l}x=4a+1\\y=4b+1\\z=4c+1\\t=4 d+1\end{array}\right.$ we have to prove that :

$x+y+z+t=8\Longrightarrow \sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}\le 6$

The function $x\mapsto \sqrt{x}$ is concave, so Jensen's inequality gives :

$\frac{\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt{t}}{4}\le \sqrt{\frac{x+y+z+t}{4}}=\sqrt{\frac{8}{4}}=\sqrt{ 2}$

Hence : $\color{red}\boxed{\sqrt{x}+\sqrt{y}+\sqrt{z}+\sqrt {t}\le 4\sqrt{2}<6}$