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}

Question

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}

Accepted Answer

∻Solution by Steps∻ ‖ step 1 ⋮ Identify the behavior of each graph to determine which represents $h$, $h'$, and $h''$. ‖ ‖ 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.⚹◊What is the relationship between the graph labeled III and the second derivative of the function $h$ in the provided scenario?⍭ Generate me a similar question◊

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