We choose $I = [-1,1]$ and write $C_{n,m}$ instead of $C_{n,m}^I$. We fix an extremal pair $(Q_0, R_0)$. Deduce from questions 4.27 and 4.29 that there exists an extremal pair $(Q_1, R_1)$ such that:
$$\left\|Q_1\right\|_I = \left|Q_1(-1)\right| \quad \text{and} \quad \left\|R_1\right\|_I = \left|R_1(1)\right|$$