Determine the maximum and the minimum of an expression

Question

Let $x,y,z \in \Bbb R, x,y,z \gt 0$ such that $x^2+y^2+z^2=1$. Determine tha maximum and the minimum possible values of the expression $$\frac {x^3+y^3+z^3} {x+y+z}.$$

Answer 1

The rearrangement inequality implies that $$3(x^3+y^3+z^3)\ge (x+y+z)(x^2+y^2+z^2).$$ So the minimum is $\frac13$, attained when $x=y=z$.

Since obviously $x,y,z\in(0,1)$, $$\frac{x^3+y^3+z^3}{x+y+z}<1\tag1.$$ On the other hand, if we set $y=z=\varepsilon$ and $x=\sqrt{1-2\varepsilon^2}$, then $$\lim_{\varepsilon\to 0}\frac{x^3+y^3+z^3}{x+y+z}=1.$$ That proves the upper bound $1$ in (1) is sharp and the said function does not have a maximum.