# Math Help - proof of inequality

1. ## proof of inequality

$please\,\, have \,\,a \,\,try !$

$(1) a+b+c=1$

$(2)a,b,c \in R^+$

$prove :\,\,27(a^3+b^3+c^3)\geq 6(a^2+b^2+c^2)+1$

$with \,\,greatest \,\, appreciation \,\,! !$

2. ## Re: proof of inequality

Originally Posted by albert940410
$(1) a+b+c=1$ $(2)a,b,c \in R^+$ $prove :\,\,27(a^3+b^3+c^3)\geq 6(a^2+b^2+c^2)+1$
One way: consider $f(x,y,z)=27(x^3+y^3+z^3)-6(x^2+y^2+z^2)-1$ defined on the compact set

$K=\{(x,y,z)\in\mathbb{R}^3:x+y+z+1=0,\;x\geq 0,\;y\geq 0,\;z\geq 0\}$

Prove that the absolute minimum of $f$ is $0$ .

3. ## Re: proof of inequality

You could use Lagrange multipliers or a similar technique, but I'm pretty sure this can be solved without calculus (e.g. Muirhead's inequality or AM-GM).