ГДЗ по алгебре 8 класс Макарычев ФГОС Задание 1026

\[\boxed{\text{1026.\ }\text{еуроки}\text{-}\text{ответы}\text{\ }\text{на}\text{\ }\text{пятёрку}}\]

\[\textbf{а)}\ ac + \frac{b}{c} \geq 2\sqrt{\text{ab}},\ \ a > 0,\ \]

\[\ b > 0,\ \ c > 0\]

\[\frac{ac^{2} + b}{c} \geq 2\sqrt{\text{ab}}\ \ \ \ | \cdot c\]

\[ac^{2} + b \geq 2c\sqrt{\text{ab}}\]

\[ac^{2} + b \geq 2 \cdot \sqrt{c^{2}\text{ab}}\]

\[\Longrightarrow ac^{2} + b \geq 2c\sqrt{\text{ab}} \Longrightarrow\]

\[\Longrightarrow верно,\ ч.т.д.\]

\[\textbf{б)}\ \left( 1 + \frac{a^{2}}{\text{bc}} \right)\left( 1 + \frac{b^{2}}{\text{ac}} \right) \cdot\]

\[\cdot \left( 1 + \frac{c^{2}}{\text{ab}} \right) \geq 8\]

\[\frac{bc + a^{2}}{\text{bc}} \cdot \frac{ac + b^{2}}{\text{ac}} \cdot \frac{ab + c^{2}}{\text{ab}} \geq 8\ \ \]

\[\ \ | \cdot a^{2}b^{2}c^{2}\]

\[\left( bc + a^{2} \right)\left( ac + b^{2} \right) \cdot\]

\[\cdot \left( ab + c^{2} \right) \geq 8a^{2}b^{2}c^{2}\]

\[\left\{ \begin{matrix} \frac{bc + a^{2}}{2} \geq \sqrt{a^{2}\text{bc}} \\ \frac{ac + b^{2}}{2} \geq \sqrt{b^{2}\text{ac}} \\ \frac{ab + c^{2}}{2} \geq \sqrt{c^{2}\text{ab}} \\ \end{matrix} \right.\ \ \ \ \ \ \ \Longrightarrow\]

\[\Longrightarrow \frac{bc + a^{2}}{2} \cdot \frac{ac + b^{2}}{2} \cdot \frac{ab + c^{2}}{2} \geq\]

\[\geq \sqrt{a^{2}\text{bc}} \cdot \sqrt{b^{2}\text{ac}} \cdot \sqrt{c^{2}\text{ab}}\ \ \ | \cdot 8\]

\[\left( bc + a^{2} \right)\left( ac + b^{2} \right)\left( ab + c^{2} \right) \geq\]

\[\geq 8a^{2}b^{2}c^{2} \Longrightarrow ч.т.д.\]