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Q.
The true set of real values of $x$ for which the function, $f(x)=x \ln x-x+1$ is positive is
Application of Derivatives
Solution:
$ f ( x )= x \ln x - x +1 \Rightarrow f ^{\prime}( x )=1+\ln x -1 $
$ \text { Hence } f ( x ) \text { is } \uparrow \text { for } x >1 \text { and } f ( x ) \text { is } \downarrow \text { for } 0< x <1$
$\Rightarrow f (1) \text { is the least value }$
$\therefore f ( x )> f (1) \text { for all } x >1 \text { as well as } 0< x <1 $
$\therefore x \ln x - x +1>0 \text { (f }(1)=0) \Rightarrow \text { (D) }$
