Question

Let $f\left(x\right)=\ln x-x+1$for $x>0,$ then which of the following is/are true :

• A. $f\left(x\right)$ has a maxima at $x=1$
• B. $f\left(x\right)$ has a minima at $x=1$
• C. $f\left(x\right)\le 0$
• D. $f\left(x\right)\ge 0$

Answer 1

Answer: (A), (C)

$f\left(x\right)\le 0$

Let $f\left(x\right)=\ln x-x+1;x\in (0,\infty )$
$\Rightarrow f^{\prime }\left(x\right)=\frac{1}{x}-1$
The derivatitve is positive for $0<x<1$ and negative for $x>1.$
Hence, function has maximum at the point $x=1$ i.e.
$\Rightarrow f\left(1\right)=\ln 1-1+1=0$
Thus for $x>0$
$\Rightarrow f\left(x\right)\le 0$