Throughout this part $A$ and $B$ denote real symmetric matrices of $\mathcal { M } _ { 2 } ( \mathbb { R } )$. We denote by $\lambda _ { 1 } \leqslant \lambda _ { 2 }$ (resp. $\mu _ { 1 } \leqslant \mu _ { 2 }$) the eigenvalues of $A$ (resp. $B$). We define $R ( A ) = \left\{ { } ^ { t } X A X \mid X \in \mathbb { R } ^ { n } , \| X \| = 1 \right\}$.
Prove that $R ( A ) = \left[ \lambda _ { 1 } , \lambda _ { 2 } \right]$.