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:

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

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