Mathematics - Application of Derivative Question with Solution | TestHub

Given $f\left(x\right)=4{\log }_e\left(x-1\right)-2x^2+4x+5,x>1$

$f^{\text{'}}\left(x\right)=\frac{4}{x-1}-4\left(x-1\right)$

For $1<x<2\Rightarrow f^{\text{'}}\left(x\right)>0$

For $x>2\Rightarrow f^{\text{'}}\left(x\right)<0$  

Hence, $f\left(x\right)$ has a maxima at $x=2$

$f\left(x\right)=-1$ has two solution as the curve of $f\left(x\right)$ will cut it at exactly two points.

Now, $f\left(e\right)>0$ and $f\left(e+1\right)<0$

i.e. $f\left(e\right)·f\left(e+1\right)<0$  

$f^{\text{'}}\left(e\right)-f^{"}\left(2\right)=\frac{4}{e-1}-4\left(e-1\right)+8>0$