= Solution
Admissible terminal wealth must lie in $(0,\infty)$, the domain of the utility; alternatively extend $U$ by $-\infty$ outside that domain. A one-period <state-price density> has $Z>0$ and $\mathbb E[ZP_1]=P_0$, componentwise. Therefore for $W=H\cdot P_1$, $\mathbb E[ZW]=H\cdot P_0=x$.
The dual definition gives the pointwise inequality $U(w)\leq\widehat U(z)+zw$ for $w,z>0$. Apply it with $z=yZ$ and take <expectations> on the finite state space:
$$
\boxed{\mathbb E U(H\cdot P_1)\leq\mathbb E\widehat U(yZ)+yx.}
$$
Equality holds precisely when $U'(w)=z$, because that is the unique maximizing payoff in the dual definition. If the feasible <portfolio> $H^*$ has $U'(H^*\cdot P_1)=y^*Z^*$, then
$$
\mathbb E U(H^*\cdot P_1)=\mathbb E\widehat U(y^*Z^*)+y^*x.
$$
The same right side bounds every competing feasible <portfolio>. Thus \b[$H^*$ is optimal.] This is the <one-period marginal-utility certificate of optimality>; it does not require the erroneous second-differentiability assertion in part (a).
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