Question

Find the intervals of increase and decrease of the, following functions.
$f\left(x\right)=(2^x-1)(2^x-2{)}^2$

Answer 1

$f\left(x\right)=(2^x-1)(2^x-2{)}^2$

$f^{\prime }\left(x\right)=2(2^x-1)(2^x-2)\left(2^x\ln 2\right)+2^x\ln 2(2^x-2{)}^2$

$=\left(2^x\ln 2\right)(2^x-2)(2^x.3-4)$

let $2^x=t$

$(t-2)(3t-4)<0$

$t=2^xϵ[\frac{4}{3},2]$

$xϵ[{\log }_2\frac{4}{3},1]$ decreasing

increasing for other values