Determine if a function is strictly increasing/decreasing

[tompenny]

Homework Statement

Determine where a function is strictly increasing/decreasing

Relevant Equations

$$f(x)=x+\frac{1}{(x+1)}$$

Hi there.

I have the following function:

$$f(x)=x+\frac{1}{(x+1)}$$

I've caculated the derivative to:

$$f'(x)=1-\frac{1}{(1+x)^2}$$

And the domain to: $$(-\infty, -1)\cup(-1, \infty)$$

I've also found two extreme point: $$x=0, x=-2$$

I know that a function is strictly increasing if:
$$f'(x)> 0$$
and strictly decreasing if:
$$f'(x)< 0$$

I've calculated the intervalls where the funtcion is strictly increasing to:
$$(-\infty, -2]\cup[0, \infty)$$
and strictly decreasing to:
$$[-2, -1)\cup(-1, 0]$$

My question is if this is correct or if the intervalls should be:
$$(-\infty, -2)\cup(0, \infty)$$ and $$(-2, -1)\cup(-1, 0)$$ instead?
As you can notice I'm very unsecure whether I should use ( or [ at the extremum points?

Any help would be greatly appreciated.
Thank you:)

http://asciimath.org/
http://docs.mathjax.org/en/v1.1-latest/tex.html#supported-latex-commands

 

[PeroK]

Can a function be strictly increasing and strictly decreasing at the same point?

 

[tompenny]

I really don't know.. are you reffering to the extremum point at x=-2?

 

[PeroK]

I really don't know.. are you reffering to the extremum point at x=-2?

Yes. In your first solution you have the two points where $f'(x) = 0$ (i.e. $x = 0$ and $x = -2$) as both strictly incraesing and strictly decreasing. That makes no sense.

Your second solution has got to be right.

 

[pasmith]

Can a function be strictly increasing and strictly decreasing at the same point?

It depends on what you mean by "strictly increasing at x".

A function is strictly increasing or decreasing on some subset of its domain. It can't be strictly increasing or decreasing on a subset consisting of a single point, because the definitions don't really make sense in that case.[1]

Here we have:

It is true that f is strictly increasing on [0,\infty).
It is true that f is strictly decreasing on (-1,0].

On the other hand, if by "f is strictly increasing at x" you mean "there is an open neighbourhood of x on which f is strictly increasing" then f is not strictly increasing or decreasing at 0.

[1]: If X consists of a single point then (\forall x \in X)(\forall y \in X) ((x &lt; y) \implies (f(x) &lt; f(y))) and (\forall x \in X)(\forall y \in X) ((x &lt; y) \implies (f(x) &gt; f(y))) are true, because whatever the choice of x and y the implications reduce to "false implies false", which is a true statement.