Using the sign chart for $$ f^{\prime \prime}(x)=-3(x+8)^{3}(x+4)^{2} $$, and only that sign chart, what conclusions can be drawn about the graph of $$ f $$ on the interval $$ x

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Using the sign chart for $$ f^{\prime \prime}(x)=-3(x+8)^{3}(x+4)^{2} $$, and only that sign chart, what conclusions can be drawn about the graph of $$ f $$ on the interval $$ x<-8 $$ ? Answer $$ \begin{array}{ll}	ext { it is positive } & 	ext { it is negative } \ 	ext { it is increasing } & 	ext { it is decreasing } \ 	ext { is concave up } & 	ext { it is concave down } \ 	ext { none of the above } & 	ext { Submit Answer }\end{array} $$

UpStudy AI · Accepted Answer

To analyze the behavior of the function $$ f $$ on the interval $$ x < -8 $$ using the second derivative $$ f''(x) = -3(x+8)^3(x+4)^2 $$, let's follow these steps: 1. **Determine the Sign of $$ f''(x) $$ for $$ x < -8 $$:** Choose a test point in the interval $$ x < -8 $$, for example, $$ x = -9 $$: $$ f''(-9) = -3(-9+8)^3(-9+4)^2 = -3(-1)^3(-5)^2 = -3(-1)(25) = 75 $$ Since $$ f''(-9) > 0 $$, $$ f''(x) $$ is positive on $$ x < -8 $$. 2. **Interpret the Sign of $$ f''(x) $$:** - **Concavity:** A positive second derivative ($$ f''(x) > 0 $$) indicates that the graph of $$ f $$ is **concave up** on this interval. - **Behavior of $$ f'(x) $$:** While $$ f''(x) > 0 $$ implies that the first derivative $$ f'(x) $$ is increasing, this information alone does not directly tell us whether $$ f(x) $$ itself is increasing or decreasing without additional data about $$ f'(x) $$. 3. **Conclusion:** Based solely on the sign chart for $$ f''(x) $$ and the fact that $$ f''(x) > 0 $$ for $$ x < -8 $$, the primary conclusion we can draw is about the concavity of $$ f $$. **Answer:** The graph of $$ f $$ is **concave up** on the interval $$ x < -8 $$. The graph of $$ f $$ is **concave up** on the interval $$ x < -8 $$.

Using the sign chart for , and only that sign chart, what
conclusions can be drawn about the graph of on the interval ?
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Using the sign chart for , and only that sign chart, what conclusions can be drawn about the graph of on the interval ? Answer