Question

Find the greatest and the least values of the following functions.
$f\left(x\right)={2.2}^{3x}-{9.2}^{2x}+{12.2}^x$ on the interval [-1, 1].

Answer 1

$f\left(x\right)=2.(2^x{)}^3-9.(2^x{)}^2+12\left(2^x\right)$

$f^{\prime }\left(x\right)=\left[6\right(2^x{)}^2-18\left(2^x\right)+12]{.2}^x.ln2$

$f^{\prime }\left(x\right)=0;6(2^x{)}^2-18(2^x)+12=0$

$\Rightarrow (2^x{)}^2-3(2^x)+2=0$

$(2^x-1)(2^x-2)=0$

$x=0,x=1\Rightarrow$ roots for maxima

$f\left(0\right)=2-9+12=5$

$f\left(1\right)=16-36+12=-8$

$f(-1)=\frac{1}{4}-\frac{9}{4}+6=4$

greatest value =$5$

least value =$-4$