ГДЗ по алгебре и начала математического анализа 11 класс Колягин Задание 287

\[\text{a\ }и\ b - искомые\ числа:\]

\[a \bullet b = 625;\ a > 0;\ \ b > 0;\]

\[b = \frac{625}{a}.\]

\[1)\ S(a) = a^{2} + b^{2} = a^{2} + \frac{625^{2}}{a^{2}};\]

\[S^{'}(a) = 2a + 625^{2} \bullet \left( - \frac{2}{a^{3}} \right) =\]

\[= 2a - \frac{2 \bullet 625^{2}}{a^{3}} = \frac{2a^{4} - 2 \bullet 625^{2}}{a^{3}}.\]

\[2)\ 2a^{4} - 2 \bullet 625^{2} \geq 0\]

\[a^{4} - 625^{2} \geq 0\]

\[\left( a^{2} + 625 \right)\left( a^{2} - 625 \right) \geq 0\]

\[(a + 25)(a - 25) \geq 0\]

\[a \leq - 25;\text{\ \ \ a} \geq 25.\]

\[3)\ a = 25;\]

\[{b = \frac{625}{25} = 25. }{Ответ:\ \ 625 = 25 \bullet 25.}\]