To determine on which intervals the function $y = h(x)$ is decreasing at an increasing rate, we need to interpret two main concepts:

• Decreasing: The function values are getting smaller as $x$ increases.
• Increasing rate: This means that the second derivative $h''(x)$ is positive, so the rate of decrease is slowing down (concave up).

From the graph, we can identify decreasing intervals by observing the downward slopes and increasing rate by checking for concavity.

Step-by-Step Analysis:

• The function is decreasing when the slope is negative (the curve is going downward).
• The function is concave up when it is shaped like a bowl (even if it's going down), indicating an increasing rate of change.

From the graph:

1. The function is decreasing from $x = -1$ to $x = 3$, since the curve slopes downward between these two points.
2. The function changes concavity (inflection points) at $x = 1$ (concave down to concave up).

Thus, the interval where the function is decreasing at an increasing rate (concave up while decreasing) is:

• From $x = 1$ to $x = 3$.

Would you like more details or have any further questions?

Here are 5 related questions to deepen your understanding:

1. How do we determine concavity from a graph using second derivatives?
2. What are the critical points, and how do they help analyze increasing or decreasing functions?
3. Can a function be increasing at a decreasing rate? How would the graph look?
4. How would you compute inflection points algebraically from a function's equation?
5. Can a function be concave up while increasing? What would this represent graphically?

Tip: When analyzing a graph, use inflection points to separate regions of concave up and concave down behavior.