= Solution
For the boundary constraint $g(x,y)=x^2+y^2-1=0$, the <Lagrange multiplier> equations for $z=xy$ are
$$
y=2\lambda x,\qquad x=2\lambda y,\qquad x^2+y^2=1.
$$
Neither coordinate can vanish at a stationary point. Substitution gives $4\lambda^2=1$. For $\lambda=1/2$, $x=y=\pm1/\sqrt2$ and $z=1/2$; for $\lambda=-1/2$, $x=-y=\pm1/\sqrt2$ and $z=-1/2$. The constraint circle is compact, its <gradient> is nowhere zero, and every extremum must occur among these stationary points. Thus
$$
\boxed{z_{\max}=\frac12\text{ at }(\pm2^{-1/2},\pm2^{-1/2}),\quad z_{\min}=-\frac12\text{ at }(\pm2^{-1/2},\mp2^{-1/2}).}
$$
The paired signs in each expression are linked. The inequality $2|xy|\le x^2+y^2=1$ also confirms the global values.
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