Exercise 2.2.15

$E=|\mathit{Ax}-b{\text{|}}^2={(C-0)}^2+{(C+D-8)}^2+{(C+3D-8)}^2+{(C+4D-20)}^2$, take derivatives w.r.t. $C$ and $D$ and let the derivatives equal to 0, we have

Divide both equations by 2, we have

Combin into matrix form, we have $\left[\begin{array}{cc}\hfill 4\hfill & \hfill 8\hfill \\ \hfill 8\hfill & \hfill 26\hfill \end{array}\right]\left[\begin{array}{c}\hfill C\hfill \\ \hfill D\hfill \end{array}\right]=\left[\begin{array}{c}\hfill 36\hfill \\ \hfill 112\hfill \end{array}\right]$, which recovers $A^{T}A\widehat{x}=A^{T}b$.