Известно, что 2<=a<=3 и 1<=b<=2. Оцените произведение (a-b)(a+b) и разность (a^2-b^2). Сравните результаты.

Ответ:

\[Дано:\ \ 2 \leq a \leq 3\ \ \ и\ \ \ 1 \leq b \leq 2.\]

\[1)\ 0 \leq (a - b)(a + b) \leq 10\]

\[- 1 \geq - b \geq - 2\]

\[+ \left| \begin{matrix} 2 \leq a \leq 3 \\ - 2 \leq b \leq - 1 \\ \end{matrix} \right.\ \]

\[\overline{\ \ \ \ \ 0 \leq a - b \leq 2}\]

\[+ \left| \begin{matrix} 2 \leq a \leq 3 \\ 1 \leq b \leq 2 \\ \end{matrix} \right.\ \]

\[\overline{\ \ \ \ 3 \leq a + b \leq 5}\]

\[х\left| \begin{matrix} 0 \leq a - b \leq 2 \\ 3 \leq a + b \leq 5 \\ \end{matrix} \right.\ \]

\[\overline{\ \ \ 0 \leq (a - b)(a + b) \leq 10.}\]

\[2)\ 0 \leq a² - b^{2} \leq 8\]

\[4 \leq a^{2} \leq 9\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ 1 \leq b^{2} \leq 4\]

\[\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ - 1 \geq - b^{2} \geq - 4\]

\[+ \left| \begin{matrix} \ 4 \leq a^{2} \leq 9 \\ - 4 \leq - b^{2} \leq - 1 \\ \end{matrix} \right.\ \]

\[\text{\ \ }\overline{\ \ \ \ \ 0 \leq a^{2} - b^{2} \leq 8.}\]