Just plug 1, 2, 3

Algebra Level 3

Let a + b + c = 1 a+b+c = 1 for reals a a , b b , and c c .

Maximize

4 a + 1 + 4 b + 1 + 4 c + 1 \large \sqrt { 4a+1 } +\sqrt { 4b+1 } +\sqrt { 4c+1 } .

Give your answer to 3 decimal places.


The answer is 4.582.

Priyanshu Mishra by Priyanshu Mishra , 20, India

This section requires Javascript.
You are seeing this because something didn't load right. We suggest you, (a) try refreshing the page, (b) enabling javascript if it is disabled on your browser and, finally, (c) loading the non-javascript version of this page . We're sorry about the hassle.

1 solution

Chew-Seong Cheong
Nov 21, 2016

By Cauchy-Schwarz inequality,

( 4 a + 1 + 4 b + 1 + 4 c + 1 ) 2 ≤ 3 ( 4 a + 1 + 4 b + 1 + 4 c + 1 ) = 21 4 a + 1 + 4 b + 1 + 4 c + 1 ≤ 21 ≈ 4.583 \begin{aligned} (\sqrt {4a+1}+\sqrt {4b+1}+\sqrt {4c+1})^2 &\le 3(4a+1+4b+1+4c+1)=21 \\ \sqrt {4a+1}+\sqrt {4b+1}+\sqrt {4c+1}&\le \sqrt{21} \approx \boxed {4.583}\end{aligned}

4 Helpful 0 Interesting 0 Brilliant 0 Confused

0 pending reports

×

Problem Loading...

Note Loading...

Set Loading...