= Solution
Put $a=f'(0)>0$. Linearizing the <morphogen reaction-diffusion equation> at $C=0$ gives
$$
C_t=DC_{xx}+aC.
$$
Seek separated modes $C(x,t)=X(x)e^{\lambda t}$. Then
$$
DX''+aX=\lambda X,
\qquad
X(0)=0,\quad X'(L)=0.
$$
The <mixed Dirichlet-Neumann modes on an interval> are
$$
X_n(x)=\sin(k_nx),
\qquad
k_n=\frac{(n+1/2)\pi}{L},
\qquad n=0,1,2,\ldots.
$$
Indeed, the Dirichlet condition selects sine functions, while the Neumann condition requires $\cos(k_nL)=0$. Their growth rates are
$$
\lambda_n=a-Dk_n^2
=a-D\frac{(n+1/2)^2\pi^2}{L^2}.
$$
The largest growth rate is the lowest mode,
$$
\lambda_0=a-\frac{D\pi^2}{4L^2}.
$$
Linear stability requires $\lambda_0<0$, after which all higher modes also decay. Therefore the <critical length for a linearly growing morphogen> is
$$
\boxed{
L<\frac\pi2\sqrt{\frac{D}{f'(0)}}.}
$$
Equality gives a neutral lowest mode, while a larger domain is linearly unstable.
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