Question

Is `f(x)=(x-3)/sqrt(x+3)` increasing or decreasing at `x=3`?

Answer 1

You have to look at the sign of the derivative of the function for `x=3`

Explanation:

`f(x) = (x-3)/sqrt(x+3) = (x-3)(x+3)^(-1/2)`

`(df)/(dx) = (x+3)^(-1/2) -1/2 (x-3)(x+3)^(-3/2)`

`(df)/(dx) |_(x=3) = 1/sqrt(6) > 0`

So the function is increasing.

On the other hand it is easy to see that `x=3` is a zero of `f(x)` and as the denominator is always positive and the numerator is negative for `x<3` and positive for `x>3` the function must be increasing.