= Solution
Let $U$ be real and twice continuously differentiable, take a nonzero real <wave number>, and write $c=c_r+ic_i$. Exponential instability requires $kc_i>0$, in particular $c_i\ne0$, so $U-c$ has no zero on the channel. Divide the <Rayleigh equation for inviscid shear flow> by $U-c$, multiply by $\psi^*$ and integrate. The wall conditions remove the boundary term, giving
$$
I+\int_{y_1}^{y_2}\frac{U''}{U-c}|\psi|^2dy=0,\qquad I=\int_{y_1}^{y_2}(|\psi'|^2+k^2|\psi|^2)dy>0.
$$
Its imaginary part is
$$
c_i\int_{y_1}^{y_2}\frac{U''|\psi|^2}{(U-c_r)^2+c_i^2}\,dy=0.
$$
The weight is nonnegative and positive wherever the nontrivial <eigenfunction> does not vanish. A nontrivial solution of this regular second-order equation cannot vanish on an interval. If $U''$ had one strict sign without changing sign, the <integral> could not vanish. If $U''$ were identically zero, the real identity would instead give the contradiction $I=0$. Thus $U''$ must take both signs, and continuity gives an interior zero. This proves \b[<Rayleigh's inflection-point theorem>: an unstable profile must have an <inflection point>], in fact a sign change of $U''$ rather than merely an isolated zero without sign change.
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