Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 28

Suppose that $f$ is a continuous function, the graph of its derivative is shown below



a.) State at what intervals is $f$ increasing or decreasing.
b.) At what values of $x$ does $f$ have a local maximum or minimum.
c.) At what intervals of $x$ is $f$ has an upward concavity.
d.) What is/are the point(s) of inflection.
e.) Sketch the graph of $f$ assuming that $f(0) = 0$

a.) Based from the graph, $f$ is increasing (where $f'$ is positive) at intervals $1 < x < 6$ and $8 < x < 9$. On the other hand, $f$ is decreasing (when $f'$ is negative) at intervals $0 < x < 1$ and $6 < x < 8$

b.) $f$ has a local maximum at $x = 6$ because at that point, $f'$ changes from positive to negative. On the other hand, $f$ has a local minima at $x = 1$ and $x = 8$ because at that points, $f'$ changes from negative to positive.

c.) $f$ has an upward concavity at intervals $0 < x < 2$, $3 < x < 5$ and $7 < x < 9$ since $f'$ is increasing there. On the other hand, $f$ has a downward concavity at intervals $2 < x < 3$ and $5 < x < 7$ since $f'$ is decreasing at these intervals.

d.) $f$ has an inflection points at $x = 2$, 3, 5 and 7 since the slope there is zero.

e.) Using the informations we obtain, the graph of $f$ might look like this