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

The correct option is B has exactly one root in $(1,3)$

$f\left(x\right)=x\log x–3+x$
$f^{\prime }\left(x\right)=1+\log x+1=2+\log x$
So $f\left(x\right)$ is monotonically increasing

We now that,
$f\left(1\right)f\left(3\right)=–2\left(3\log 3\right)<0$
By using intermmediate value theorem,
Hence, one root must lie in $(1,3)$.