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Write $s=S_0$, assume the usual nonzero initial price vector, and defineThe covariance matrix $V$ is positive definite. If $s,\mu$ are linearly independent, then $D>0$ by the strict Cauchy-Schwarz inequality. The Lagrangian for mean-variance optimization can be writtenIts first-order equation gives $\pi =\lambda V^{-1}s+\eta V^{-1}\mu$. The two constraints then giveStrict convexity makes this the unique constrained minimizer. Its variance isThe mean-variance efficient frontier is the upper branch $m\ge Bx/A$: points on the lower branch have a larger-mean competitor of the same variance. The picture shows the generic nondegenerate case at a fixed initial budget. With only two stocks and independent budget and mean constraints, every feasible portfolio lies on the curve; additional independent portfolio directions allow larger variances at the same mean.

If $\mu =\kappa s$, then $D=0$, and only $m=\kappa x$ is feasible. The variance minimizer is $(x/A)V^{-1}s$, still in the stated span; no two-dimensional frontier exists. This degeneracy is not excluded merely by nonsingularity of $V$.

For exponential utility and Gaussian prices, the moment-generating function givesMaximizing this is equivalent to maximizing ${\mu }^T\pi -\gamma {\pi }^{T}V\pi /2$ subject to $s^T\pi =y$. Its strictly concave first-order equation givesThus the same two funds suffice.

For the Gaussian two-fund theorem with general utility, the Gaussian assumption and the exponential growth bounds justify the differentiations and expectations. Gaussian integration by parts applied to $S_1=\mu +V^{1/2}Z$ givesAt any finite maximizer the left side is a multiplier $\eta s$. If $U$ is concave and non-affine, and $\pi \ne 0$, then ${\pi }^T{S}_1$ is a nondegenerate Gaussian with positive density everywhere. Since continuous $U^{\prime \prime }\le 0$ is strictly negative on some interval, its expectation is strictly negative. Rearranging provesThe zero portfolio already belongs to that span.

There is a genuine qualification in the final printed assertion: increasing and concave does not exclude affine utility. For $U(w)=w$, $s=\mu =(1,1,1{)}^T$, and $V=I$, every budget-feasible portfolio is optimal. With $y=1$, $(1,0,0{)}^T$ is optimal but is outside the common line spanned by ${\pi }_A,{\pi }_B$. If $\mu$ is not proportional to $s$, affine utility has no maximizer at all, because a zero-cost direction with positive mean can be scaled without bound. Thus the universal claim about the maximizing portfolio needs non-affine utility and existence of a finite maximizer. Under the literal weaker assumptions there is still a two-fund optimum whenever an optimum exists: the affine constant-mean case permits choosing the minimum-variance portfolio.