Single Variable Calculus, Chapter 4, 4.3, Section 4.3, Problem 8

Below is the graph of $f'$


a.) At what intervals is $f$ increasing and decreasing? Explain.
b.) State and explain at what values of $x$ does $f$ have a local maximum or minimum.
c.) State and explain at what intervals does $f$ has upward concavity or downward concavity.
d.) What are the $x$=coordinates of the inflection points of $f$? Why?

a.) $f$ is increasing (when $f'$ is positive) at intervals $2 < x < 4$ and $6 < x < 9$. On the other hand, $f$ is decreasing (when $f'$ is negative) at intervals $0 < x < 2$ and $4 < x < 6$.

b.) $f$ has a local minimum at $x = 2$ and $x = 6$ (assuming that they have the same slope) because at these points; $f'(x)$ changes from negative to positive. On the other hand, $f$ has a local maximum at $x = 4$ because at that point, $f'(x)$ changes from positive to negative.

c.) $f$ has an upward concavity at intervals $1 < x < 3$, $5 < x < 7$ and $8 < x < 9 $ because $f'(x)$ is inreasing at these intervals. On the other hand, $f$ has a downard concavity at intervals $0 < x < 1$, $3 < x < 5$ and $ 7 < x < 8$ because $f'(x)$ is decreasing at these intervals.

d.) $f$ has an inflection points at $x = 1$, $x = 3$, $x = 5$, $x = 7$ and $x = 8$ because at these points, the slope of $f'$ is zero.