Strategies to prove inequalities with interval notation How to prove a inequalities with interval notation, for example:

Find minimum of $a^3+b^3+c^3$ with $a,b,c \in [-1;\infty), a^2+b^2+c^2=9$


By Holder: if $a,b,c>0$
$$(a^3+b^3+c^3)(a^3+b^3+c^3)(1+1+1)\ge (a^2+b^2+c^2)^3$$
then 
$$a^3+b^3+c^3\ge 9\sqrt{3}$$
and when $a=b=c=\sqrt{3}$
case 2:let $a\ge b\ge 0>c\ge -1$
then $a^2+b^2>8$,then $a>2sqrt{2}$,so $a^3+b^3+c^3>a^3+c^3>a^3-1>\sqrt{512}-1>9\sqrt{3}$
case 3:let $a>0>b>c\ge -1$,and we have $a^2>7$,so $a>\sqrt{7}$,then
$a^3+b^3+c^3>\sqrt{343}-2>9\sqrt{3}$
