Question

determine os valores de k para que a função f(x)=(2-3k)^x seja crescente e decrescente? me ajudeeem por favor!

Answer 1

Para função exponencial temos:

$f(x) = a^x$

Para função crescente, temos $a\ \textgreater \ 1$:

$2 - 3k \ \textgreater \ 1 \\ \\ -3k \ \textgreater \ -1 \\ \\ 3k \ \textless \ 1 \\ \\ \boxed{k\ \textless \ \frac{1}{3}}$

Para função decrescente, temos $0\ \textless \ a\ \textless \ 1$:

$0\ \textless \ 2-3k\ \textless \ 1 \\ \\ \left \{ {{2-3k\ \textgreater \ 0} \atop {2-3k\ \textless \ 1}} \right. \\ \\ 2 - 3k\ \textgreater \ 0 \to -3k \ \textgreater \ -2 \to 3k\ \textless \ 2 \to \boxed{k\ \textless \ \frac{2}{3} } \\ \\ 2-3k\ \textless \ 1 \to -3k\ \textless \ -1 \to 3k\ \textgreater \ 1 \to\boxed{ k\ \textgreater \ \frac{1}{3} } \\ \\ \boxed{ \frac{1}{3} \ \textless \ k\ \textless \ \frac{2}{3} }$