Question

Is `f(x)=sqrt(xe^x+3x)` increasing or decreasing at `x=0`?

Answer 1

I would argue that a reasonable definition of "increasing at a point", should probably say that this function is increasing at `0`.

Explanation:

If the definition of "`f` is increasing at `a`" is `f'(a) > 0`, then this function is not increasing at `0` because `f'(0)` is not defined.

However:
`f(0) = 0` and for all other `x` in the domain of `f`, we have `f(x) > 0`. So I think we should say that `f` is increasing at `0`.