Докажите неравенство a^2-8ab+17b^2-2b+3>0.

Ответ:

\[a^{2} - 8ab + 17b^{2} - 2b + 3 > 0\]

\[\left( a^{2} - 8ab + 16b^{2} \right) + \left( b^{2} - 2b + 1 \right) + 2 > 0\]

\[\underset{\geq 0}{\overset{(a - 4b)^{2}}{︸}} + \underset{\geq 0}{\overset{(b - 1)^{2}}{︸}}\underset{> 0}{\overset{+ 2}{︸}} > 0 \Longrightarrow верно.\]