Prove the following saddle-node bifurcation theorem

Question

Prove the following saddle-node bifurcation theorem:

Suppose that $f_λ$ depends smoothly on the parameter $λ$ and satisfies:

(a) $f_{λ_0}(x_0)=x_0$

(b) $f_{λ_0}'(x_0)=1$

(c) $f_{λ_0}''(x_0) \ne 0$

(d) $\frac{\partial f_λ}{\partial λ} \big|_{λ=λ_0}(x_0) \ne 0$

Then there is an interval Ι about $x_0$ and a smooth function $\mu:Ι\to R$ satisfying $\mu(x_0)=λ_0$ and such that $f_\mu(x) (x)=x$