# Homework Help: Help with finding first derivative and critical points?

1. Nov 13, 2012

### Brianna V

1. The problem statement, all variables and given/known data
What are the critical points of function f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 ?

2. Relevant equations
When f'(x) equals 0 or is undefined, x is a critical number.

3. The attempt at a solution

The original function is f(x) = 2 (x^2 + 4)^(1/2) - 4x + 24 .
I got the derivative as f'(x) = [2x / (x^2 + 4)^(1/2)] - 4 .
What are the critical points? x = 0 is the only critical point I figure since 2x = 0, x = 0 (setting the numerator equal to zero). Any confirmation here?

Or do I need to put everything under a common denominator and figure stuff out that way?
i.e. 2x/[(x^2+4)^(1/2)] - 4[(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
= 2x - 4 /[(x^2+4)^(1/2)]
= 2x - 4 /[(x^2+4)^(1/2)] * [(x^2+4)^(1/2)]/[(x^2+4)^(1/2)]
= 2x[(x^2+4)^(1/2)]-4(x^2+4) /x^2+4
= 2x[(x^2+4)^(1/2)]-4x^2-16 /x^2+4
...But now what :S....

Last edited: Nov 14, 2012
2. Nov 13, 2012

### Ray Vickson

The function has no critical points: it is strictly decreasing on ℝ. You can see this by writing f'(x) with a common denominator. You tried this, but you wrote it incorrectly on your second line above: you wrote
$$2x - \frac{4}{(x^2+4)^{1/2}},$$
but possibly you meant
$$\frac{2x-4}{(x^2+4)^{1/2}},$$
which would still be wrong. I suggest you start again, and be careful.

RGV