= Solution
For $a=b=1$, the <stationary points> found in parts (a) and (b) have <temperatures> $0$, $1$, $1/4$ and $1/5$. The <unit sphere> is compact and $T$ is a <continuous function>, so the <extreme value theorem> ensures that a hottest <point> exists; it must be one of those <stationary points>. Thus
$$
\boxed{T_{\max}=1,\qquad (x,y,z)=(1,0,0)\text{ or }(-1,0,0).}
$$
For a direct bound, put $s=x^2\in[0,1]$. The identity $(y^2-z^2)^2\geq0$ gives $y^2z^2\leq(y^2+z^2)^2/4=(1-s)^2/4$, and
$$
T\leq s^2+\frac{(1-s)^2}{4}\leq1,
$$
since $1-s^2-(1-s)^2/4=(1-s)(5s+3)/4\geq0$. Equality forces $s=1$, giving exactly the two <points> above.
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