= Solution
Using the rate parameter convention for the <exponential distribution>, <independence> gives the <joint probability density function>
$$
\boxed{f_{U,V}(u,v)=2e^{-u-2v}\mathbf1_{\{u\geq0,\ v\geq0\}}.}
$$
For $z\geq0$, the survival <probability> of the minimum is
$$
\mathbb P(Z>z)=\mathbb P(U>z,V>z)=e^{-z}e^{-2z}=e^{-3z}.
$$
Thus \b[$Z$ has exponential distribution with rate $3$], and
$$
f_Z(z)=3e^{-3z}\mathbf1_{\{z\geq0\}}.
$$
Values of a <probability density function> at the single endpoint $0$ do not affect the <probability distribution>.
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