Question

Math

Posted 6 months ago

```
Let $h$ be a twice differentiable function except at a vertical asymptote. One of these graphs is the graph of $h$, one is of $h^{\prime}$ and one is of $h^{\prime \prime}$.
Choose the option that matches each function with its appropriate graph.
Choose 1 answer:
(A) \begin{tabular}{|llll} \hline $\mathrm{A}$ & $h$ & $h^{\prime}$ & $h^{\prime \prime}$ \\ \hline & I & II & III \end{tabular}
(B) \begin{tabular}{|llll} \hline $\mathrm{B}$ & $h$ & $h^{\prime}$ & $h^{\prime \prime}$ \\ \hline & I & III & II \\ \hline \end{tabular}
(C) \begin{tabular}{llll} $\mathrm{C}$ & $h$ & $h^{\prime}$ & $h^{\prime \prime}$ \\ \hline & II & III & I \end{tabular}
(D) \begin{tabular}{|llll|}
\hline $\mathrm{D}$ & $h$ & $h^{\prime}$ & $h^{\prime \prime}$ \\
\hline & III & II & $\mathrm{I}$ \\
\hline
\end{tabular}
```

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Answer from Sia

Posted 6 months ago

Solution by Steps

step 2

Graph I: The blue curve crosses the y-axis and x-axis, indicating it is a function with a root and changes in concavity. This suggests it could be $h$

step 3

Graph II: The orange curve approaches the x-axis as $x$ approaches infinity, indicating it is a function that decreases and levels off. This suggests it could be $h'$

step 4

Graph III: The purple curve has a vertical asymptote at $x=1$ and changes direction sharply, indicating it is a function with a discontinuity in its derivative. This suggests it could be $h''$

step 5

Match the graphs to the functions based on their behavior: $h$ is Graph I, $h'$ is Graph II, and $h''$ is Graph III

A

Key Concept

Graph behavior and function matching

Explanation

By analyzing the behavior of each graph, we can determine which graph represents the function, its first derivative, and its second derivative.

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