Solution:

We need to carefully analyze the given properties of $f$ (decreasing, negative, and concave down) and use them to determine the behaviors of $f'$ (the first derivative) and $f''$ (the second derivative).

1. Behavior of $f'$:

• Since $f$ is decreasing, the slope of $f$ is negative. This means $f' < 0$.
• Additionally, since $f$ is concave down, the rate of change of $f'$ (i.e., $f''$) is negative, so $f'$ is decreasing.

Thus, the graph of $f'$ is negative and decreasing.
Answer: The graph of $f'$ would be negative and decreasing because $f$ is decreasing and concave down.

2. Behavior of $f''$:

• Since $f$ is concave down, the second derivative $f'' < 0$. This means the graph of $f''$ is negative only.

Answer: The graph of $f''$ would be negative only because $f$ is concave down.

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Final Responses:

• $f'$: Negative and decreasing because $f$ is decreasing and concave down.
• $f''$: Negative only because $f$ is concave down.

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Would you like to discuss this further or need more examples? Here are five related questions:

1. What would happen to $f'$ and $f''$ if $f$ were concave up instead of concave down?
2. How can the sign of $f''$ help identify intervals of concavity for $f$?
3. If $f$ is increasing and concave down, what would $f'$ and $f''$ look like?
4. Why is $f'$ negative and decreasing if $f$ is both decreasing and concave down?
5. How does the concavity of $f$ affect the rate of change of $f'$?

Tip: Always check the sign and behavior (increasing/decreasing) of derivatives to infer properties of the original function!