Answer
There is at least one root in the specified interval.
Work Step by Step
$f(x) = \ln x - x + \sqrt x$
Now use the intervals $(2, 3)$:
$f(2) = \ln 2 - 2 + \sqrt 2$
$f(2) = -1.307 + 1.414$
$f(2) \approx 0.107$
$f(3) = \ln 3 - 3 + \sqrt 3$
$f(2) = -1.901 + 1.732$
$f(2) \approx -0.169$
So by definition of the Intermediate Value Theorem, $f(x)$ will take all values between $0.107$ and $-0.169$. So there is at least one root/value of $x$ in $(2,3)$ for which $f(x)$ is zero.
