Question

Consider a function $f(x)=x e^x+\left(x e^x\right)^{-1}$ where $x \in R-\{0\}$, then which of the following statement ( s ) is/are INCORRECT?

• A. $f(x)$ attains its local minima at $x=-1$.
• B. $f(x)$ attains its local maxima at $x=\beta$, where $\beta \in(0,1)$.
• C. $f(x)$ attains its local minima at $x=\beta$, where $\beta \in(0,1)$.
• D. $f(x)$ attains its local minima at $x=\beta$, where $\beta \in(1, \infty)$.

Answer 1

Answer: (A), (B), (D)

$ f ( x )= xe ^{ x }+\frac{1}{ xe ^{ x }}$ where $x \neq 0$.
$f ^{\prime}( x )=\left( xe ^{ x }+ e ^{ x }\right)-\frac{1}{\left( xe ^{ x }\right)^2}\left( xe ^{ x }+ e ^{ x }\right)=\frac{\left( xe ^{ x }+ e ^{ x }\right)\left( xe ^{ x }-1\right)\left( xe ^{ x }+1\right)}{\left( xe ^{ x }\right)^2} $
$=\frac{ e ^{3 x }( x +1)\left( x + e ^{- x }\right)\left( x - e ^{- x }\right)}{\left( xe ^{ x }\right)^2} $
$f ^{\prime}( x )=0 ; x =-1, e ^{- x }=- x \text { (No solution) } $
$e ^{- x }= x (\text { exactly one solution, let it is } \beta)$
$\text { at } x =-1, \text { local maxima } $
$\text { at } x =\beta \in(0,1) \text { local minima. }$