Rearranging part i and using the polarization identity gives
$$
\begin{aligned}
t[f(x_{k+1})-f(u)]
&\leq(x_k-x_{k+1})^T(x_{k+1}-u)\\
&=\frac12\left(\|u-x_k\|^2-\|u-x_{k+1}\|^2-\|x_k-x_{k+1}\|^2\right).
\end{aligned}
$$
Dropping the final nonpositive term proves
$$
\boxed{t[f(x_{k+1})-f(u)]
\leq\frac12(\|u-x_k\|^2-\|u-x_{k+1}\|^2).}
$$


Solution

Past exam of the mathematics course of the University of Cambridge / 2022 / iii / Paper 339 / 1 / b / ii / Solution

 Solution

ID: past-exam-of-the-mathematics-course-of-the-university-of-cambridge/2022/iii/paper-339/1/b/ii/solution