Answer
$f^{'}(x)= 68.265 [1+20.5e^{-0.9x}]^{-2} e^{-0.9x} $
$f^{''}(x)= 2518.9785[1+20.5e^{-0.9x}]^{-3} [e^{-0.9x}]^2- 61.4385 [1+20.5e^{-0.9x}]^{-2} \times e^{-0.9x} $
POINT OF INFLECTION
$x=\frac{\ln 20.5}{0.9} \approx3.356$[correct to three decimal places]
Work Step by Step
$f(x)=\frac{3.7}{1+20.5e^{-0.9x} }$
$f(x)=3.7[1+20.5e^{-0.9x}]^{-1}$
Taking derivative with respect to x
$f^{'}(x)= 3.7(-1) [1+20.5e^{-0.9x}]^{-2} \times 20.5e^{-0.9x}\times (-0.9) $
$f^{'}(x)= 68.265 [1+20.5e^{-0.9x}]^{-2} e^{-0.9x} $
Taking derivative again
$f^{''}(x)= 68.265(-2) [1+20.5e^{-0.9x}]^{-3} \times20.5 e^{-0.9x}\times(-0.9)\times e^{-0.9x}+ 68.265 [1+20.5e^{-0.9x}]^{-2} \times e^{-0.9x}\times (-0.9) $
$f^{''}(x)= 2518.9785[1+20.5e^{-0.9x}]^{-3} [e^{-0.9x}]^2- 61.4385 [1+20.5e^{-0.9x}]^{-2} \times e^{-0.9x} $
For point of inflection
Put
$f^{''}(x)= 2518.9785[1+20.5e^{-0.9x}]^{-3} [e^{-0.9x}]^2- 61.4385 [1+20.5e^{-0.9x}]^{-2} \times e^{-0.9x} =0$
$ 2518.9785[1+20.5e^{-0.9x}]^{-3} [e^{-0.9x}]^2= 61.4385 [1+20.5e^{-0.9x}]^{-2} \times e^{-0.9x} $
$ 2518.9785[1+20.5e^{-0.9x}]^{-1} [e^{-0.9x}]^1= 61.4385 $
$[ 1+20.5e^{-0.9x}]^{-1} [e^{-0.9x}]^1= \frac{ 61.4385 }{ 2518.9785 } $
$\frac{1}{ [ 1+20.5e^{-0.9x}] e^{0.9x}] }=\frac{ 61.4385 }{ 2518.9785 } $
$\frac{ 2518.9785 }{ 61.4385 }= [ 1+20.5e^{-0.9x}] e^{0.9x}] $
$\frac{ 2518.9785 }{ 61.4385 }= e^{0.9x} +20.5 $
$e^{0.9x} +20.5 =41$
$e^{0.9x} =41-20.5=20.5$
$\ln e^{0.9x} =\ln 20.5$
$ 0.9x\ln e =\ln 20.5$
$ 0.9x =\ln 20.5$
$x=\frac{\ln 20.5}{0.9} \approx3.356$[correct to three decimal places]
