ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 1006

\[\boxed{\text{1006.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]

\[a > 0,\ \ b > 0\]

\[\textbf{а)}\ (a + b)(ab + 16) \geq 16ab\]

\[\frac{a + b}{2} \geq \sqrt{\text{ab}},\ \ \]

\[\frac{ab + 16}{2} \geq \sqrt{16ab}\]

\[(a + b) \geq 2\sqrt{\text{ab}},\]

\[\ \ ab + 16 \geq 8\sqrt{\text{ab}}\]

\[\Longrightarrow (a + b)(ab + 16) \geq\]

\[\geq 2\sqrt{\text{ab}} \cdot 8\sqrt{\text{ab}}\]

\[(a + b)(ab + 16) \geq\]

\[\geq 16ab \Longrightarrow ч.т.д.\]

\[\textbf{б)}\ \left( a^{2} + 4b \right)(4b + 25) \geq 80ab\]

\[\frac{a^{2} + 4b}{2} \geq \sqrt{a^{2} \cdot 4b},\ \ \]

\[a^{2} + 4b \geq 4a\sqrt{b}\]

\[\frac{4b + 25}{2} \geq \sqrt{4b \cdot 25},\]

\[\ \ 4b + 25 \geq 20\sqrt{b}\]

\[\Longrightarrow \left( a^{2} + 4b \right)(4b + 25) \geq\]

\[\geq 4a\sqrt{b} \cdot 20\sqrt{b}\]

\[\left( a^{2} + 4b \right)(4b + 25) \geq\]

\[\geq 80ab \Longrightarrow ч.т.д.\]