Question

the derivative of the twice - differentiable function ( f ) is shown below on the domain ( (-9,9) ). the graph of ( f ) has points of inflection at ( x = - 6,x=-4,x = - 1,x = 5 ), indicated by small green circles. what inferences can be made about the graphs of ( f,f ), and ( f ) on the interval ( (-6,-4) )? choose the best answer for each dropdown.

answer attempt 1 out of 2

from the figure given above, it can be seen that the graph of ( f ) on the interval ( (-6,-4) ) is (_), and (_).

based on these observations, it can be concluded that:

on the interval ( (-6,-4) ), the graph of ( f ) would be (_) because ( f ) is (_).

on the interval ( (-6,-4) ), the graph of ( f ) would be (_) because ( f ) is (_).

Explanation:

Step1: Analyze $f'$ on $(-6,-4)$

Looking at the graph of $f'$, on $(-6,-4)$:

• $f'(x) > 0$ (lies above x-axis)
• $f'(x)$ is increasing (rising from left to right)
• $f'(x)$ is concave down (curving downward, since the inflection points are at $x=-6$ and $x=-4$, so between them the graph of $f'$ has downward concavity)

Step2: Relate $f'$ to $f$

For $f$: When $f'(x) > 0$, $f$ is increasing. When $f'$ is increasing, $f''(x) > 0$, so $f$ is concave up.

Step3: Relate $f'$ to $f''$

For $f''$: $f''$ is the derivative of $f'$. When $f'$ is concave down, its derivative $f''$ is decreasing. Also, since $f'$ is increasing, $f''(x) > 0$.

Answer:

From the figure given above, it can be seen that the graph of $f'$ on the interval $(-6,-4)$ is positive, increasing, and concave down.

Based on these observations, it can be concluded that:
On the interval $(-6,-4)$, the graph of $f$ would be increasing and concave up because $f'$ is positive and increasing.
On the interval $(-6,-4)$, the graph of $f''$ would be positive and decreasing because $f'$ is increasing and concave down.