Expert: Alon Mandes - 5/19/2009

Question
For this problem,
f(x)=      6x^2
      -----------
        x^2+16

The two q's below are the two I can't figure out.

(G) Use interval notation to indicate where f(x) is concave down.


(H) List the x values of all inflection points of f. If there are no inflection points, enter 'NONE'.




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This problem is similar, which I am having the same difficulties.

 f(x)=     6x
       -----------
         x^2-16

I have been trying forever and can't seem to get these 3,

b)Use interval notation to indicate where f(x) is decreasing.

(F) Use interval notation to indicate where f(x) is concave up.

(G) Use interval notation to indicate where f(x) is concave down.

For some reason I never can get the concave stuff, I know you find f''(x). Then use the 1st derivative test. I just don't seem to get the right answer.


***************
THIS AGAIN IS LIKE THE OTHER TWO!!!
I Just NEED HELP FIGURING OUT THESE SIMILAR PROBLEMS:

f(x)=    6
     --------
      x^2-16

b) Use interval notation to indicate where f(x) is increasing.

c) Use interval notation to indicate where f(x) is decreasing.

f) Use interval notation to indicate where f(x) is concave up.

g) Use interval notation to indicate where f(x) is concave down.



(I really just need help explaining these, I know what I need to do to find the answers, but I never get them right. I have no idea where I go wrong, but I know you are busy so any help really does HELP!)


Answer
f(x)=6x²/(x²+16) .
f'(x)=12x/(x²+16) - 70x³/((x²+16)²
Let's find extremum points : we solve f'(x)=0 :
12x/(x²+16) - 70x³/((x²+16)² = 0
12x/(x²+16) = 70x³/((x²+16)²
12x(x²+16) = 70x³
x³+16x=6x³
x(16-5x²)=0 The solutions are x=0 & x=4/√5 .

Increasing Interval : f'(x)>0 : 12x/(x²+16) > 70x³/((x²+16)²
                               12x(x²+16) > 70x³
                               x³+16x > 6x³
                               x(16-5x²)>0
                               Solutions :
                               0<x<4/√5 & x<-4/√5 .

Increasing Interval : f'(x)<0 : 12x/(x²+16) < 70x³/((x²+16)²
                               12x(x²+16) < 70x³
                               x³+16x < 6x³
                               x(16-5x²)<0
                               Solutions :
                               -4√5<x<0 & x>4/√5 .


You may continue & do the same the rest of the functions.
Any problem , I'm here .

Alon.