Let's go through each part step-by-step, based on the graph of $f'(x)$ (the derivative of $f(x)$) provided.

Analyzing the Graph of $f'(x)$:

1. Identify intervals where $f(x)$ is increasing or decreasing:

   • When $f'(x) > 0$, $f(x)$ is increasing.
   • When $f'(x) < 0$, $f(x)$ is decreasing.

2. Identify concavity of $f(x)$:

   • When $f'(x)$ is increasing (i.e., $f''(x) > 0$), $f(x)$ is concave up.
   • When $f'(x)$ is decreasing (i.e., $f''(x) < 0$), $f(x)$ is concave down.

3. Local Minima of $f(x)$:

   • Local minima of $f(x)$ occur where $f'(x)$ changes from negative to positive.

4. Inflection Points of $f(x)$:

   • Inflection points of $f(x)$ occur where $f'(x)$ changes concavity, which corresponds to local extrema (peaks and troughs) of $f'(x)$.

Solutions

Part (A): For what values of $x$ in $(0,8)$ is $f(x)$ increasing?

The function $f(x)$ is increasing where $f'(x) > 0$:

• From approximately $x \in (1, 2.5) \cup (5, 7)$.

Answer: $(1, 2.5) \cup (5, 7)$

Part (B): For what values of $x$ in $(0,8)$ is $f(x)$ concave down?

The function $f(x)$ is concave down where $f'(x)$ is decreasing (slopes downward):

• From approximately $x \in (0, 1) \cup (3, 5) \cup (6, 8)$.

Answer: $(0, 1) \cup (3, 5) \cup (6, 8)$

Part (C): Find all values of $x$ in $(0,8)$ where $f(x)$ has a local minimum.

Local minima of $f(x)$ occur where $f'(x)$ changes from negative to positive:

• At approximately $x = 2.5$ and $x = 7$.

Local Minima: $2.5, 7$

Part (D): Find all values of $x$ in $(0,8)$ where $f(x)$ has an inflection point.

Inflection points occur at the local extrema of $f'(x)$:

• At approximately $x = 1, 3, 5,$ and $6$.

Inflection Points: $1, 3, 5, 6$

Would you like further details or explanations on any of these solutions?

Related Questions:

1. How can we determine intervals of concavity from the second derivative?
2. What indicates a local maximum versus a local minimum in the context of $f'(x)$?
3. Why does a change in concavity signify an inflection point?
4. How can we determine intervals where $f(x)$ is decreasing?
5. What is the significance of the points where $f'(x) = 0$?

Tip:

In analyzing the graph of $f'(x)$, always look for where it crosses the x-axis, changes slope direction, and has peaks or valleys to deduce properties about $f(x)$.