Proof that if $x$ minimises $\left \| Ax-b \right \|^{2}_2 +\mu\left \| x \right \|^{2}_2$ then it solves regularised normal equations

Question

I'm looking for a proof that if $x$ minimises $\left \| Ax-b \right \|^{2}_2 +\mu\left \| x \right \|^{2}_2$ then it solves $(A^{T}A+\mu I)x=A^{T}b$