# Absolute Extrema Problem Stuck again

• Mar 23rd 2009, 05:59 AM
Jim Marnell
Absolute Extrema Problem Stuck again
I though i had a grasp of these but this one is a little different from the other ones.

Find the locations (ordered pairs) of all absolute extrema (min max) for the following function with the specified domain.

$\displaystyle F(x)=\frac{x}{x^2+2}$ Domain [0,4]

Heres what ive done so far:
$\displaystyle f'(x)=\frac{(x^2+2)(1)-(x)(2x)}{(x^2+2)^2}$
$\displaystyle F'(x)=\frac{(x^2+2)+(2x^2)}{(x^2+2)^2}$
$\displaystyle F'(x)=\frac{3x^2+2}{(x^2+2)^2}$

The bottom can never equal 0 so how do i test to find the absolute max and absolute min? Thanks for any help!
• Mar 23rd 2009, 06:22 AM
Pinkk
You suddenly switched signs in the numerator. This is what you should have:

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

$\displaystyle f'(x)=0$ when $\displaystyle 2-x^{2}=0$ and that happens when $\displaystyle x=\pm \sqrt{2}$. However, $\displaystyle x=-\sqrt{2}$ is not in the interval $\displaystyle [0,4]$, so the only critical point to be concerned with is $\displaystyle x=\sqrt{2}$. Evaluate $\displaystyle f(0),f(\sqrt{2}),f(4)$ and determine which are your max and min.
• Mar 23rd 2009, 06:40 AM
HallsofIvy
Quote:

Originally Posted by Jim Marnell
I though i had a grasp of these but this one is a little different from the other ones.

Find the locations (ordered pairs) of all absolute extrema (min max) for the following function with the specified domain.

$\displaystyle F(x)=\frac{x}{x^2+2}$ Domain [0,4]

Heres what ive done so far:
$\displaystyle f'(x)=\frac{(x^2+2)(1)-(x)(2x)}{(x^2+2)^2}$
$\displaystyle F'(x)=\frac{(x^2+2)+(2x^2)}{(x^2+2)^2}$

What happened to the "-"?
That should be
$\displaystyle f'(x)= \frac{(x^2+ 2)- (2x^2)}{x^2+ 2)^2}$

Quote:

$\displaystyle F'(x)=\frac{3x^2+2}{(x^2+2)^2}$

The bottom can never equal 0 so how do i test to find the absolute max and absolute min? Thanks for any help!
• Mar 23rd 2009, 08:04 AM
Jim Marnell
f(0)=0
f($\displaystyle \sqrt{2}$=0
f(4)=$\displaystyle \frac{-7}{162}$

Did i do this wrong, cause those numbers seem weird. If not would max be 0,0 and min be $\displaystyle (4,\frac{-7}{162})$
• Mar 23rd 2009, 08:45 AM
Pinkk
You have to plug those values into $\displaystyle f(x)$, not $\displaystyle f'(x)$

$\displaystyle f(0)=0$
$\displaystyle f(\sqrt{2})=\frac{\sqrt{2}}{4}$
$\displaystyle f(4)=\frac{2}{9}$

Your maximum is at $\displaystyle (\sqrt{2},\frac{\sqrt{2}}{4})$ and your minimum is at $\displaystyle (0,0)$.