Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 326 / 2 / b / iv / Solution

The equation $(A^{\ast }A+{\alpha }_{\ast }I)u_{\ast }=A^{\ast }f$ is the Tikhonov normal equation. For $u=u_{\ast }+h$, expand the Tikhonov regularization functional:The normal equation makes the linear term zero. The final two terms are nonnegative and are strictly positive for $h\ne 0$ because ${\alpha }_{\ast }>0$. Thus $u_{\ast }$ is the unique global minimizer.