Let's analyze the graph and solve each part of the question.

The graph shown is of $y = f'(x)$, the derivative of the function $f(x)$. Here's the reasoning for each question:

(a) On what intervals is $f(x)$ increasing?

• $f(x)$ increases when $f'(x) > 0$ (above the x-axis).
• From the graph:
  • $f'(x) > 0$ on $(0, 2) \cup (4, 6)$.

Answer: $(0, 2) \cup (4, 6)$

(b) On what intervals is $f(x)$ decreasing?

• $f(x)$ decreases when $f'(x) < 0$ (below the x-axis).
• From the graph:
  • $f'(x) < 0$ on $(2, 4) \cup (6, \infty)$.

Answer: $(2, 4) \cup (6, \infty)$

(c) At what value(s) of $x$ does $f(x)$ have a local maximum?

• A local maximum occurs when $f'(x)$ changes from positive to negative (crosses the x-axis from above to below).
• From the graph:
  • $f'(x)$ changes from positive to negative at $x = 2$.

Answer: $2$

(d) On what interval(s) is $f(x)$ concave upward?

• $f(x)$ is concave upward when $f''(x) > 0$, which occurs when $f'(x)$ is increasing.
• From the graph:
  • $f'(x)$ increases on $(3, 5)$.

Answer: $(3, 5)$

Let me know if you want further clarification or detailed steps for any part!

Related Questions:

1. How can we identify concavity using the second derivative of $f(x)$?
2. Why does the sign of $f'(x)$ determine whether $f(x)$ is increasing or decreasing?
3. What happens to $f(x)$ at points where $f'(x) = 0$?
4. How do we interpret the graph of $f''(x)$ if it were provided instead of $f'(x)$?
5. Can $f(x)$ have a point of inflection if $f'(x)$ doesn't cross the x-axis?

Tip:

Always examine where the derivative $f'(x)$ is zero or undefined to locate critical points for $f(x)$.