= Solution
For every deflator $Y$ and $\varepsilon\geq0$, $Y^*+\varepsilon Y$ is a deflator. Minimality and right differentiation at zero give
$$
\mathbb E[\widehat u'(Y_1^*)Y_1]+X_0Y_0\geq0.
$$
Taking $Y=Y^*$ and varying the positive scalar multiple $(1+\varepsilon)Y^*$ in both directions around one gives equality.
Set $\xi_1=-\widehat u'(Y_1^*)$ and $\xi_0=X_0$. The preceding inequality says
$$
\mathbb E[\xi_1Y_1]\leq X_0Y_0
$$
for every deflator, with equality at $Y^*$. Part c produces $H^*$ with $H^*\cdot P_0=X_0$ and $H^*\cdot P_1=\xi_1$. The inverse relation between conjugate derivatives gives $u'(\xi_1)=Y_1^*$, so equality holds in part d:
$$
\mathbb E[u(H^*\cdot P_1)]
=\mathbb E[\widehat u(Y_1^*)]+X_0Y_0^*.
$$
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