Traveling Wave Solution of Bistable Diffusion Equation

Question

I would like some help understanding a proof regarding the existence of traveling wave solutions to the equation $$u_t = u_{xx} + f(u),$$ where $f(u)$ is assumed to be continuously differentiable. We seek solutions of the form $u(x,t) = v(x-ct) = v(\eta)$ that tend to different constant values, i.e $\lim_{x \to \pm \infty} u(x,t) = a^{\pm}$. Substituting $v(\eta)$ into the PDE, we arrive at the system of first order equations \begin{align*} \dot{v}_1 &= v_2 \\ \dot{v}_2 &= -f(v_1)-cv_2 \end{align*} Since $\lim_{\eta \to \pm \infty} v = a^{\pm}$, we expect $a^{\pm}$ to be fixed points of the system. This implies that $f(a^{\pm}) = 0$. If we calculate the Jacobian of the system evaluated at one of these fixed points, we find that the eigenvalues satisfy $$\lambda = \frac{-c \pm \sqrt{c^2 - 4f'(a^{\pm})}}{2}. $$ The textbook I'm reading asserts that in order for the traveling wave solution to exist, we need $f'(a^{\pm}) \leq 0$, so that each fixed point is a saddle. I understand that two different saddle points can connect trajectories, but why can't say $(a^{-},0)$ be an unstable node and $(a^{+},0)$ be a stable node? Thanks in advance!