Finding critical numbers for f(x) = 2x/√(x²+5)

[cal.queen92]

Homework Statement

I need to find the derivative of f(x) = (2x)/(sqrt((x^2)+5) and then simplify it in order to find the critical numbers of the derivative and find the intervals of increase and decrease of the function. (I am doing some curve sketching and keep getting lost!)

The Attempt at a Solution

I end up with a derivative like this:

((2)(x^2 +5)^(1/2) - (2x)(x)(x^2 + 5)^(-1/2)) / (sqrt(x^2 +5))^2

Now, I need to simplify the answer, so I factored out: 2(x^2 +5)^(-1/2) and got an answer that looks like this:

5/ (x^2 +5)(sqrt(x^2 +5))

So now I'm stuck! With this derivative, I'm trying to find the intervals of increase and decrease of the function, but I'm not sure how to find the critical numbers of this derivative.

I have a suspicion that I am not factoring out the exponents properly. Is that the case? If not, am i going wrong at all? How do I find the critical numbers of a derivative that looks like this?

Help!

Thank you so much.

 

[fzero]

You missed a factor of 2,

$$f'(x) =\frac{ 10}{ (x^2 +5)(\sqrt{x^2 +5})} = \frac{ 10}{ (x^2 +5)^{3/2}}.$$

Now the critical points of $$f(x)$$ are the points where $$f'(x)=0$$. However in this case, there are no points where this occurs, though $$f'(x)\rightarrow 0$$ as $$x\rightarrow \pm \infty$$. You should still be able to sketch $$f'(x)$$ to determine the behavior of the slope of $$f(x)$$.