\[\boxed{\text{849.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]
\[\textbf{а)}\ \frac{c^{2} + 1}{2} \geq c\]
\[\frac{c^{2} + 1}{2} - c \geq 0\]
\[\frac{(c - 1)^{2}}{2} \geq 0\]
\[верно,\ так\ как\ (c - 1)^{2} \geq 0.\]
\[\textbf{б)}\ \frac{c}{c^{2} + 1} \leq \frac{1}{2}\]
\[\frac{c}{c^{2} + 1} - \frac{1}{2} \leq 0\]
\[\frac{c^{2} - 2c + 1}{2 \cdot \left( c^{2} + 1 \right)} \geq 0\]
\[\frac{(c - 1)^{2}}{2 \cdot \left( c^{2} + 1 \right)} \geq 0\]
\[верно,\ так\ как\ (c - 1)^{2} > 0,\ \ \]
\[\left( c^{2} + 1 \right) > 0,\ \ 2 > 0.\]