Question

The equation $x\log x=3-x$

• A. has no root in $(1,3)$
• B. has exactly one root in $(1,3)$
• C. $x\log x-(3-x)>0$ in $[1,3]$
• D. $x\log x-(3-x)<0$ in $[1,3]$

Answer 1

Answer: (B)

has exactly one root in $(1,3)$

$f\left(x\right)=x\log x-3+x$
$f^{\prime }\left(x\right)=1+\log x+1$
$f\left(1\right)f\left(3\right)=-2\left(3\log 3\right)=-ve$
Hence one roots in $(1,3)$