I can now view the graphs. Let's break down each part and determine the intervals where the functions are increasing or decreasing.

a) First graph:

• The graph shows a parabola-like shape.
• Increasing: The function is increasing after the local minimum (around $x = 1$).
• Decreasing: The function is decreasing before the local minimum, to the left of $x = 1$.

Answer:

• Increasing: $(1, \infty)$
• Decreasing: $(-\infty, 1)$

b) Second graph:

• The function has an inflection point and then grows rapidly.
• Increasing: The function increases everywhere after $x = 0$.
• Decreasing: No decreasing interval is observed.

Answer:

• Increasing: $(0, \infty)$
• Decreasing: NONE

c) Third graph:

• The graph appears to be sinusoidal, with alternating increasing and decreasing intervals.
• Increasing: The function increases on the intervals $(-1, 0)$ and $(2, 3)$.
• Decreasing: The function decreases on the intervals $(0, 2)$ and $(3, 4)$.

Answer:

• Increasing: $(-1, 0) \cup (2, 3)$
• Decreasing: $(0, 2) \cup (3, 4)$

Would you like further details or explanations on any of the intervals? Here are some related questions to expand:

1. How can you determine increasing or decreasing behavior from the derivative?
2. What role do critical points play in analyzing a function’s intervals?
3. How does concavity affect the increasing/decreasing nature of a function?
4. What is the significance of inflection points in determining function behavior?
5. Can a function have more than one increasing and decreasing interval?

Tip: Use the first derivative of a function to confirm increasing or decreasing intervals by checking its sign (positive for increasing, negative for decreasing).