Concavity, inflection ps, intervals of F.

[phillyolly]

Homework Statement

(a) Find the intervals on which is increasing or decreasing.
(b) Find the local maximum and minimum values of .
(c) Find the intervals of concavity and the inflection points.

F=x²/(x²+3

The Attempt at a Solution

a) f ' =6x/(x²+3)²

6x=0 => x=0

What concluusion can I draw from this data about increase/decrease?
I am asking because my function actually does not go below y=0, so I thought that it does not decrease at all. My answer is that f is only increasing on intervals for which f'(x) > 0.

b) local minimum is at (0,0). no local maximum.
c) inflection points: x=+-1.

Stuck with concavity!

 

[Bohrok]

So setting the first derivative equal to 0 gave you a local max or min; finding the sign of f'' at x = 0 will tell you which one it is.

Now you need to find f'' and set it equal to 0 to find the concavity next. The points of inflection are where the concavities change; with two points of inflection, x = -1 and x = 1, you have three intervals on which to check the sign of f'' to determine the concavity of f on each of those intervals.

 

[Dick]

This is the same question as https://www.physicsforums.com/showthread.php?t=415080 Don't repost, phillyolly, if you're not willing to follow up on the first post.

 

[phillyolly]

I am sorry about that. I forgot about that post, thank you for reminding.

 

[Dick]

I am sorry about that. I forgot about that post, thank you for reminding.

Ok, so given Bohrok's hint, what are the intervals of concavity? Where is f'' positive and negative?

 

[phillyolly]

My problem is that I cannot tell the difference between the domain, concavity, increase and decrease. There are different formulas, different approaches. I have been searching for two days, contacting my friends, looking through math forums.

If someone can just show me how to do this one problem, I will do the next ten on my own in my homework, please.

 

[Bohrok]

You don't need to consider the domain of any functions to find concavity.

The three intervals I was talking about are (-∞, -1), (-1, 1), (1, ∞). Each one has a concavity, and to find them, you look at whether f'' is positive or negative in each interval.
For (-1, 1), f'' is greater than 0, which means that f is concave up on that interval. If f'' is less than 0, f is concave down on that interval.
Can you finish the rest?

 

[phillyolly]

That is awesome. Finally, after so many sleepless days and nights, I completed one of my problems. Thank you a lot for your wonderful support, people of PF.