For a positive real number \(r\), let \(M(r)\) be the largest possible value of \( F(x,y,z) = x^2 +yz^3 \) over all ordered triples of non-negative real numbers \((x,y,z),\) such that \(x^2+y^2+z^2=\sqrt{r}.\)
There is a value of \(r,\) denoted by \( r^*\), such that \(F(x,y,z)=M(r^*)\) for two different triples \((x,y,z)\) which satisfy \(x^2+y^2+z^2=\sqrt{r^*}.\) If \( r^* = \frac{a}{b} \) where \(a\) and \(b\) are coprime positive integers, what is \(a + b \)?
by
Calvin Lin
