Solve the questions about the function

Question

Suppose that 

f(x)= (3x)/(x^2-49). 

A.List all critical numbers of f. If there are no critical numbers, enter 'NONE'. 

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

C. List the x-values of all local maxima of f . If there are no local maxima, enter 'NONE'. 

D. List the x-values of all local minima of f. If there are no local minima, enter 'NONE'. 

E.  List the x-values of all inflection points of f . If there are no inflection points, enter 'NONE'. 

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

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

H.List all horizontal asymptotes of f . If there are no horizontal asymptotes, enter 'NONE'. 

I. List all vertical asymptotes of f . 

Answer 1

This is a very thorough list of function characteristics.  Take a look at the answer I posted to one of your other questions for details, but here is a high level way of getting started:

 

Critical Numbers:  Set f'(x) = 0 and solve for x, set f''(x) = 0 and solve for x, check for values of x for which f(x) is undefined.

 

Increasing/Decreasing:  f(x) is increasing when f'(x) > 0, decreasing when f'(x) < 0

 

Local max/min:  Find x where f'(x) = 0 and pick values to the left and right.  A switch from + to - indicates a max, a switch from - to + indicates a min.

 

Inflection:  Find x where f''(x) = 0 and pick values to the left and right.  A switch from + to - or - to + indicates a point of inflection.

 

Concavity:  f(x) is concave up when f''(x) > 0, concave down when f''(x) < 0

 

Horizontal Asymptotes: One approach is to take the limit as x->inf and x-->-inf of the function

 

Vertical Asymptotes:  Find the values of x for which f(x) is undefined.

 

Please ask any follow-up questions you need; it would be helpful to know what places in particular that you are struggling with.

 

J.T.