Absolute Maximum

[Emenjivar22]

i have to find the aboslute maximum value of the function

f(x)= (1 over 1 + |x|) plus (1 over 1+|x+2|)

i hope that is understandable, im not sure how to start this question. this is my frist time in a calculus course. any help would be apprectiated
thanks

 

[tkhunny]

Ponder This:

For x > 0, |x| = x and |x+2| = x+2
For -2 < x < 0, |x| = -x and |x+2| = x+2
For x < -2, |x| = -x and |x+2| = -x-2

Don't forget to check the boundaries where the derivative doesn't exist.

 

[Emenjivar22]

i know i need to get the first derivative of this function in order to test the points but how to i get a derivative when there is an absolute value involved?

 

[tkhunny]

Your question is EXACTLY what I addressed. I'll make it more obvious then you can give it a go.

You have: \(\displaystyle f(x) = \frac{1}{1+|x|}+\frac{1}{1+|x+2|}\)

I'm telling your that:

If you look at only x > 0, you can work with this function: \(\displaystyle f(x) = \frac{1}{1+x}+\frac{1}{1+x+2}\)


If you look at only -2 > x > 0, you can work with this function: \(\displaystyle f(x) = \frac{1}{1-x}+\frac{1}{1+x+2}\)


If you look at only -2 > x, you can work with this function: \(\displaystyle f(x) = \frac{1}{1-x}+\frac{1}{1-x-2}\)


Think really hard on why this is so.


You are also responsible for looking at x = 0 and x = -2. The derivative will not help you, there.