Classical Inequalities addicts can do this. Part 6

Algebra Level 3

For $x, y, z> 0$ and $x^3 + y^3 + z^3 = 1$ , determine the value of

$\large \big|\max(27xyz + 26x^3 - 100)\big|$

Notation: $|\cdot|$ denotes the absolute value function.

For more problems like this, try answering this set .

The answer is 73.

1 solution

Christian Daang
Jul 23, 2017

By AM-GM,

$x^3 + 27y^3 + 27z^3 \ge 27xyz \\ \implies 27x^3 + 27y^3 + 27z^3 \ge 27xyz + 26x^3 \\ \implies \text{abs}( \max(27xyz + 26x^3 - 100)) = \text{abs}(27(x^3 + y^3 + z^3) - 100) = \boxed{73}$

Classical Inequalities addicts can do this. Part 6

The answer is 73.

1 solution

0 pending reports