Consider the function $$f:(0,\infty)\to(-\infty,\infty)$$ given by

$$f(x)=\sqrt{x}\,\log_e(x)-x+1.$$

Then which one of the following statements is TRUE?

A

The derivative of the function $$f$$ is decreasing in the interval $$(0,1)$$

B

The function $$f$$ has a local maximum at some point $$a\in(0,\infty)$$

C

The function $$f$$ has a local minimum at some point $$b\in(0,\infty)$$

D

The function $$f$$ has NEITHER a point of local maximum NOR a point of local minimum in the interval $$(0,\infty)$$