Questions: Given the graph of f'(x) below, determine the intervals on which f(x) is increasing, decreasing, concave up, and concave down. Find all minimums, maximums, and inflection points.

Given the graph of f'(x) below, determine the intervals on which f(x) is increasing, decreasing, concave up, and concave down. Find all minimums, maximums, and inflection points.

Transcript text: Given the graph of $f^{\prime}(x)$ below, determine the intervals on which $f(x)$ is increasing, decreasing, concave up, and concave down. Find all minimums, maximums, and inflection points.

Solution

Solution Steps

Step 1: Find intervals where $f(x)$ is increasing and decreasing.

$f(x)$ is increasing when $f'(x) > 0$ and decreasing when $f'(x) < 0$. From the graph of $f'(x)$, we can see that $f'(x) > 0$ on $(0,3)$ and $(6,9)$. Also, $f'(x) < 0$ on $(3,6)$.

Thus, $f(x)$ is increasing on $(0, 3)$ and $(6, 9)$, and decreasing on $(3, 6)$.

Step 2: Find minimums and maximums.

Since $f'(x)$ changes from positive to negative at $x=3$, $f(x)$ has a local maximum at $x=3$. Since $f'(x)$ changes from negative to positive at $x=6$, $f(x)$ has a local minimum at $x=6$.

Step 3: Find intervals of concavity and inflection points.

$f(x)$ is concave up when $f''(x) > 0$, which means $f'(x)$ is increasing. $f(x)$ is concave down when $f''(x) < 0$, which means $f'(x)$ is decreasing.

$f'(x)$ is increasing on the intervals $(1,5)$ and $(8,9)$. $f'(x)$ is decreasing on the intervals $(0,1)$, $(5,8)$.

Thus, $f(x)$ is concave up on $(1, 5)$ and $(8,9)$, and $f(x)$ is concave down on $(0,1)$ and $(5,8)$. Inflection points occur when the concavity changes. Thus, the inflection points of $f(x)$ are at $x = 1$, $x = 5$ and $x = 8$.