Hello everyone,(adsbygoogle = window.adsbygoogle || []).push({});

I'm back again....haha. I was just checking over my homework assignment again and realized that I'm pretty unsure of one of the assigned problems. Here it is:

Find the Intervals of Increase and Decrease, local max and min values and the concavity of the function f(x)= (x^2)/(x^2+3)

First to find the intervals of increase and decrease as well as the local max and min values I found the first derivative of the function:

f'(x)= [(x^2+3)(x)-(x^2)(2x)]/_x^2+3)^2

f ' (x) = (6x)/(x^2+3)^2

If you find where x equals zero you get:

6x=0 therefore x=0

x^2+3=0 therefore x=squareroot (-3) Now what I did was I went on to make a chart showing where the function was increasing and decreasing which I am unable to show on the computer and from that I got that it was increasing on the interval (0,infinity) and decreasing on the interval (-infinity,0). MY problem with this is that I just realized that to do this I took the square root of a negative number, which of course you cannot do. So do I just say that this is an unreal answer?

Then using the chart I made I said that a local minimum occurs at (0,0) and that a local maximum does not occur for this graph.

Then to find the concavity I found the second derivative of the function:

f''(x)= [(x^2+3)^2(6)-24x^2(x^2+3)]/(x^2+3)^4

f ''(x) = [-18x(x-1)]/(x^2+3)^3

Then once again setting x=0 you get x=0, x=1, and then you run into the taking the square root of a negative number again. Which I never noticed at the time and I showed it as +square root 3 and -square root 3. Then I made another chart and showed where the graph was concave up or down and where the inflection points were. When I did this I got it was:

Concave down (-infinity,0) U (1,infinity)

Concave Up (0,1)

Inflection points at x=0 and x=1.

I thought I did it right until I was looking through my answers just now and noticed I was taking the root of a negative number, and now I'm not really sure what to do. Thanks in advance for any advice.

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Intervals of Increase/Decrease, and concavity

**Physics Forums | Science Articles, Homework Help, Discussion**