Answer
a. f(x) is increasing in the interval $(\frac{3}{2},+\infty)$
b. f(x) is decreasing in the interval $(-\infty,\frac{3}{2})$
c. f(x) is concave up in the interval $(-\infty,\infty)$
d. f(x) has no inflection points.
Work Step by Step
In order to see if f(x) is increasing or decreasing, we must find the critical points, or when f'(x) is 0. After we find the critical points, we must find which intervals f'(x) is positive or negative in, hence the number line. f(x) is increasing when f'(x) is positive, and f(x) is decreasing when f'(x) is negative. In order to find the inflection points, we must find where f''(x) is equal to 0. In order to determine where f(x) is concave up or down, we can either find where f'(x) is increasing or decreasing or directly observe the graph of f(x) and see where the slopes are gradually increasing or decreasing.