= Solution
Put $D=b^TV^{-1}b$ and decompose any $\theta$ in the $V$-inner product as
$$
\theta=\lambda\theta_M+\eta,\qquad \eta^TV\theta_M=\eta^Tb=0.
$$
Then
$$
\mathbb EX=\lambda D,\qquad
\operatorname{Var}(X)=\lambda^2D+\eta^TV\eta.
$$
For fixed $\lambda$, a nonzero $\eta$ leaves the mean unchanged and strictly raises variance, so strict monotonicity of $F$ excludes it from a maximizer. If $D>0$ and $\lambda<0$, the zero portfolio has a larger mean and smaller variance, so this is also impossible. Hence every maximizer is
$$
\theta^*=\lambda\theta_M\qquad(\lambda\geq0).
$$
If $b=0$, the unique variance-minimizing maximizer is $\theta^*=0$, which is the same conclusion with $\lambda=0$.
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