= Solution
Let $G=x^2/a^2+y^2/a^2+z^2/b^2-1$. Its <gradient> cannot vanish on the <ellipsoid>, since the origin is not on the surface. Thus the <Lagrange multiplier> equations capture every constrained <stationary point>:
$$
\boxed{4x^3=\frac{2\lambda x}{a^2},\qquad 2yz^2=\frac{2\lambda y}{a^2},\qquad 2y^2z=\frac{2\lambda z}{b^2},\qquad \frac{x^2+y^2}{a^2}+\frac{z^2}{b^2}=1.}
$$
The <temperature> is a <sum> of squares, so $T\geq0$. Equality requires $x=0$ and $yz=0$. The surface constraint then gives the four coldest <points>:
$$
\boxed{T_{\min}=0,\qquad (x,y,z)=(0,\pm a,0)\text{ or }(0,0,\pm b).}
$$
These are all such <points>, because a <sum> of nonnegative terms can vanish only when each term vanishes.
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