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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 $\mathbb{E}[Z{P}_1]=P_0$, componentwise. Therefore for $W=H\cdot P_1$, $\mathbb{E}[ZW]=H\cdot P_0=x$.

The dual definition gives the pointwise inequality $U(w)\le \hat{U}(z)+zw$ for $w,z>0$. Apply it with $z=yZ$ and take expectations on the finite state space:Equality holds precisely when $U^{\prime }(w)=z$, because that is the unique maximizing payoff in the dual definition. If the feasible portfolio $H^{\ast }$ has $U^{\prime }(H^{\ast }\cdot P_1)=y^{\ast }{Z}^{\ast }$, thenThe same right side bounds every competing feasible portfolio. Thus $H^{\ast }$ is optimal. This is the one-period marginal-utility certificate of optimality; it does not require the erroneous second-differentiability assertion in part (a).