009A Sample Final 2, Problem 10

Let

$f(x)={\frac {4x}{x^{2}+1}}$

(a) Find all local maximum and local minimum values of  $f,$  find all intervals where  $f$  is increasing and all intervals where  $f$  is decreasing.

(b) Find all inflection points of the function  $f,$  find all intervals where the function  $f$  is concave upward and all intervals where  $f$  is concave downward.

(c) Find all horizontal asymptotes of the graph  $y=f(x).$

(d) Sketch the graph of  $y=f(x).$

Foundations:

Solution:

(a)

Step 1:

Step 2:

(b)

Step 1:

Step 2:

(c)

Step 1:

Step 2:

(d):

Final Answer:
(a)
(b)
(c)
(d)

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