Solution
= Solution
Choose two inputs $0<a<b$. Their proposed <kernel matrix> is
$$
K=\begin{pmatrix}a&b\\b&b\end{pmatrix}.
$$
The coefficient <vector> $\alpha=(1,-1)^T$ gives
$$
\alpha^TK\alpha=a-b<0.
$$
Thus $K$ is not a <positive semidefinite matrix>, and
$$
\boxed{k_2(x,x')=\max(x,x')\text{ is not a positive-semidefinite kernel}}.
$$
Equivalently, its <determinant> $ab-b^2=b(a-b)$ is negative.