Function D(x) has interval of increase when 0<x<4 because the dolphin's height is going up between 0 and 4 seconds. The interval of decrease of D(x) is 4<x<7 because the height of the dolphin is going down between 4 seconds and 7 seconds(when the dolphins jump is over). What are the intervals of increase and decrease for D(x)? what do they tell us about the path of the bird?

Based on your description of the function \( D(x) \), here are the conclusions about its intervals of increase and decrease:

1. Interval of Increase: \( 0 < x < 4 \)

   • This tells us that the dolphin's height is increasing during this time period. The dolphin is rising or jumping higher up in the air until the 4-second mark.

2. Interval of Decrease: \( 4 < x < 7 \)

   • This indicates that after reaching its peak height at 4 seconds, the dolphin's height begins to decrease until the 7-second mark, which implies the dolphin is coming back down to the water's surface.

Interpretation About the Path of the Bird

While you haven't provided any information about the bird or its path, a few inferences can be made if we were to assume any relationship:

• If the bird is observing or following the dolphin, its path could be influenced by the dolphin's movement. For instance, if the dolphin jumps high (increasing height), the bird might need to fly higher or soar above the dolphin.
• During the intervals of increase, the bird might be hopeful for an upcoming spectacle; it could hover or rise alongside the dolphin.
• Conversely, during the interval of decrease, the bird might prepare to either descend or adjust its flight path to account for the dolphin’s return to water.

Without more context about the bird (like its movement patterns or relationship to the dolphin), these interpretations remain speculative. However, they suggest a close connection between the dolphin's jumping behavior and the bird’s potential flight behavior.