Concavity, inflection ps, intervals of F.

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In summary, we have found the intervals on which the function f(x)=x^2/(x^2+3) is increasing or decreasing, the local maximum and minimum values, and the intervals of concavity and inflection points. We have also determined that the function is only increasing on intervals for which its derivative is greater than 0, and that the points of inflection are at x=-1 and x=1. The function is concave up on the interval (-1, 1) and concave down on the intervals (-∞, -1) and (1, ∞).
  • #1
phillyolly
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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!
 
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  • #2
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.
 
  • #3
Last edited:
  • #4
I am sorry about that. I forgot about that post, thank you for reminding.
 
  • #5
phillyolly said:
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?
 
  • #6
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.
 
  • #7
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?
 
  • #8
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.
 

FAQ: Concavity, inflection ps, intervals of F.

1. What is concavity and how is it determined?

Concavity is a measure of the curvature of a graph. It is determined by the sign of the second derivative of a function. If the second derivative is positive, the graph is concave up, and if it is negative, the graph is concave down.

2. What is an inflection point and how is it related to concavity?

An inflection point is a point on a graph where the concavity changes. It can be determined by finding where the second derivative changes sign. If the second derivative changes from positive to negative, the point is a local maximum and if it changes from negative to positive, the point is a local minimum.

3. How can you use the first derivative test to find intervals of increasing and decreasing functions?

The first derivative test involves finding critical points (where the first derivative is equal to 0) and evaluating the first derivative at points on either side of the critical point. If the derivative is positive on one side and negative on the other, the critical point is a local maximum. If the derivative is negative on one side and positive on the other, the critical point is a local minimum. Intervals of increasing and decreasing functions can be determined by looking at the signs of the derivative on either side of the critical points.

4. How is the second derivative used to determine the concavity of a function?

The second derivative is used because it represents the rate of change of the slope of the original function. If the second derivative is positive, it means the slope is increasing and the graph is concave up. If the second derivative is negative, it means the slope is decreasing and the graph is concave down.

5. Can a function have multiple inflection points?

Yes, a function can have multiple inflection points because the concavity can change multiple times within its domain. These points can be found by setting the second derivative equal to 0 and solving for the corresponding x-values.

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