Max and Minimum Values?

Question

Carry out the following steps to sketch the graph of
f(x) = x / (1 + x^2) :
(a) Find the local maxima and minima of f. Compute the local maximum and minimum values. Give the intervals of increase and decrease.
(b) Find the infection points of f. Give the intervals where f is concave upward and
where f is concave downward.
(c) Determine if f is even or odd.
(d) Find lim as x approaches infinity of f(x) and lim as x approaches infinity of f(x).
(e) Make a careful sketch of the graph of f that reflects the above information.

Answer 1

a. f(-1) = -0.5, f(1) = 0.5
...The function decreases from -∞ to -1, increase from -1 to +1, and decreases again from +1 to +∞.

b. The inflection foint is at f(0) = 0. Below this, the function is concave up and below this it is concave down.

c. f is odd. f(-x) = -f(x)

d. Lim f(x) as x approaches both -∞ and +∞ is 0.

e. You'll have to do that.