= Solution
Reciprocation reverses the order of positive numbers, so
$$
\boxed{k_3(x,x')=\frac1{\max(x,x')}=\min\left(\frac1x,\frac1{x'}\right)=k_1(1/x,1/x')}.
$$
This is the pullback of the <positive-semidefinite kernel> from part (i) by $x\mapsto1/x$. Explicitly, the <feature map> into the same <L2 space> is
$$
\psi(x)(t)=\mathbf1_{(0,1/x]}(t),\qquad \langle\psi(x),\psi(x')\rangle=\frac1{\max(x,x')}.
$$
Hence \b[$k_3$ is a <positive-semidefinite kernel>], with $\|\psi(x)\|_2^2=1/x<\infty$ for every allowed input.
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