= Solution
{c}
Let $F=x+y+z^2$ and $g=x^2+y^2+z^2$. The <unit sphere> is <compact>, so the <continuous function> $F$ has a <global maximum> and <global minimum>. Since $\nabla g\ne0$ on the <unit sphere>, all such points satisfy the <Lagrange multiplier> equations
$$
1=2\lambda x,\qquad1=2\lambda y,\qquad2z=2\lambda z,\qquad x^2+y^2+z^2=1.
$$
Thus $x=y$ and either $z=0$ or $\lambda=1$.
If $z=0$, then $x=y=\pm1/\sqrt2$, giving $F=\pm\sqrt2$. If $\lambda=1$, then $x=y=1/2$ and $z=\pm1/\sqrt2$, giving $F=3/2$. These exhaust the candidates, so
$$
\boxed{\min F=-\sqrt2\text{ at }(-1/\sqrt2,-1/\sqrt2,0),\qquad
\max F=\frac32\text{ at }(1/2,1/2,\pm1/\sqrt2).}
$$
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