<s>[INST] Let $x, y, z$ be real numbers. If $x + y + z = 1$ and $x^2 + y^2 + z^2 = 1$, then what is the minimum value of $x^3 + y^3 + z^3$? Through manipulation, I got as far as $x^3 +y^3 + z^3 = 1 + 3xyz$
I tried solving it using AM - GM inequality but it only gave me the maximum value.
GM - HM does not to help either.
 [/INST] You can use Cauchy-Schwarz inequality. $X+y=1-z$, thus $(1-z)^2 = (x+y)^2$ is less than or equal to $(1+1)(x^2+y^2)=2 (1-z^2)$. I get $z$ more than or equal to $-1/3$. $x^3+y^3+z^3= (x^3+y^3)+z^3=(x+y)(x^2-xy+y^2)+z^3= (3(x^2+y^2)-(x+y)^2)/2 + z^3$ doing the necessary substitutions I get $1-3 z^2 +3 z^3$, equals $5/9$.
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