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A strictly increasing differentiable concave utility function on all of $\mathbb{R}$ has $U^{\prime }(x)>0$ everywhere. Otherwise monotonicity and the decreasing derivative would make it constant on a right half-line. Its tangent at zero also shows $U(x)\to -\infty$ as $x\to -\infty$.

If $X$ has only one sign, the stated second alternative already holds. Otherwise both $\mathbb{P}(X<0)$ and $\mathbb{P}(X>0)$ are positive. As $\theta \to \infty$, $-U(\theta X)$ tends to infinity on $\{X<0\}$; as $\theta \to -\infty$, it does so on $\{X>0\}$. Since $-U\ge 0$, Fatou lemma implies $F(\theta )\to -\infty$ at both ends. Continuity gives a finite maximizer ${\theta }_{\ast }$, and differentiability gives $F^{\prime }({\theta }_{\ast })=0$.

For optimal marginal utility as a one-period pricing density, put $Z=U^{\prime }({\theta }_{\ast }X)$. It is strictly positive. On $\{|X|\ge 1\}$ it is bounded by $|X|U^{\prime }({\theta }_{\ast }X)$, whose expectation is finite by part ii. On $\{|X|<1\}$ continuity bounds $U^{\prime }$ on the compact interval $[-|{\theta }_{\ast }|,|{\theta }_{\ast }|]$. Thus $Z$ is integrable and $XZ$ is absolutely integrable. Consequently $\boxed{\mathbb{E}[XZ]=0}$. Normalizing $Z$ by its positive expectation, if desired, makes it a probability density with the same zero pricing expectation.