QuestionGiven the function $q(x)$ with values: $q(1)=-10$ , $q(3)=-6$ , $q(5)=-3$ , $q(7)=-1$ , $q(9)=0$ . Is it concave up, down, or neither?

Studdy Solution

STEP 1

Assumptions1. The function $q(x)$ is represented by the given table of values. . We are asked to determine if the function could be concave up, concave down, or neither.
3. We assume that the function is differentiable and continuous.

STEP 2

To determine the concavity of a function, we need to look at the second derivative of the function. However, we only have a table of values and not the actual function. So, we will look at the differences between successive function values, which is similar to looking at the derivative.

STEP 3

Let's calculate the differences between successive function values. This will give us an idea of how the function is changing.
\begin{tabular}{|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ \\ \hline1 & -10 & - \\ \hline3 & -6 & $-6 - (-10)$ \\ \hline5 & -3 & $-3 - (-6)$ \\ \hline7 & -1 & $-1 - (-3)$ \\ \hline9 &0 & $0 - (-1)$ \\ \hline\end{tabular}

STEP 4

Calculate the differences.
\begin{tabular}{|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ \\ \hline1 & -10 & - \\ \hline3 & -6 &4 \\ \hline & -3 &3 \\ \hline7 & -1 &2 \\ \hline9 &0 &1 \\ \hline\end{tabular}

STEP 5

Now, let's calculate the differences between these differences. This is similar to looking at the second derivative.
\begin{tabular}{|c|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ & $\Delta^2 q(x)$ \\ \hline1 & -10 & - & - \\ \hline3 & - &4 & - \\ \hline5 & -3 &3 & $3 -4$ \\ \hline7 & -1 &2 & $2 -3$ \\ \hline9 &0 &1 & $1 -2$ \\ \hline\end{tabular}

STEP 6

Calculate the second differences.
\begin{tabular}{|c|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ & $\Delta^2 q(x)$ \\ \hline1 & -10 & - & - \\ \hline3 & -6 &4 & - \\ \hline5 & -3 &3 & -1 \\ \hline & -1 &2 & -1 \\ \hline9 &0 &1 & -1 \\ \hline\end{tabular}

STEP 7

The second differences are all negative. This suggests that the function is concave down because a negative second derivative indicates a concave down function.
Therefore, the function $q(x)$ could be concave down.

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QuestionGiven the function $q(x)$ with values: $q(1)=-10$ , $q(3)=-6$ , $q(5)=-3$ , $q(7)=-1$ , $q(9)=0$ . Is it concave up, down, or neither?

Studdy Solution

STEP 1

Assumptions1. The function $q(x)$ is represented by the given table of values. . We are asked to determine if the function could be concave up, concave down, or neither.
3. We assume that the function is differentiable and continuous.

STEP 2

To determine the concavity of a function, we need to look at the second derivative of the function. However, we only have a table of values and not the actual function. So, we will look at the differences between successive function values, which is similar to looking at the derivative.

STEP 3

STEP 4

Calculate the differences.
\begin{tabular}{|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ \\ \hline1 & -10 & - \\ \hline3 & -6 &4 \\ \hline & -3 &3 \\ \hline7 & -1 &2 \\ \hline9 &0 &1 \\ \hline\end{tabular}

STEP 5

STEP 6

Calculate the second differences.
\begin{tabular}{|c|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ & $\Delta^2 q(x)$ \\ \hline1 & -10 & - & - \\ \hline3 & -6 &4 & - \\ \hline5 & -3 &3 & -1 \\ \hline & -1 &2 & -1 \\ \hline9 &0 &1 & -1 \\ \hline\end{tabular}

STEP 7

The second differences are all negative. This suggests that the function is concave down because a negative second derivative indicates a concave down function.
Therefore, the function $q(x)$ could be concave down.

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QuestionGiven the function q(x)q(x)q(x) with values: q(1)=−10q(1)=-10q(1)=−10, q(3)=−6q(3)=-6q(3)=−6, q(5)=−3q(5)=-3q(5)=−3, q(7)=−1q(7)=-1q(7)=−1, q(9)=0q(9)=0q(9)=0. Is it concave up, down, or neither?

Studdy Solution

STEP 1

Assumptions1. The function q(x)q(x)q(x) is represented by the given table of values. . We are asked to determine if the function could be concave up, concave down, or neither.3. We assume that the function is differentiable and continuous.

STEP 2

To determine the concavity of a function, we need to look at the second derivative of the function. However, we only have a table of values and not the actual function. So, we will look at the differences between successive function values, which is similar to looking at the derivative.

STEP 3

STEP 4

Calculate the differences.\begin{tabular}{|c|c|c|} \hlinexxx & q(x)q(x)q(x) & Δq(x)\Delta q(x)Δq(x) \\ \hline1 & -10 & - \\ \hline3 & -6 &4 \\ \hline & -3 &3 \\ \hline7 & -1 &2 \\ \hline9 &0 &1 \\ \hline\end{tabular}

STEP 5

STEP 6

Calculate the second differences.\begin{tabular}{|c|c|c|c|} \hlinexxx & q(x)q(x)q(x) & Δq(x)\Delta q(x)Δq(x) & Δ2q(x)\Delta^2 q(x)Δ2q(x) \\ \hline1 & -10 & - & - \\ \hline3 & -6 &4 & - \\ \hline5 & -3 &3 & -1 \\ \hline & -1 &2 & -1 \\ \hline9 &0 &1 & -1 \\ \hline\end{tabular}

STEP 7

The second differences are all negative. This suggests that the function is concave down because a negative second derivative indicates a concave down function.Therefore, the function q(x)q(x)q(x) could be concave down.

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QuestionGiven the function $q(x)$ with values: $q(1)=-10$ , $q(3)=-6$ , $q(5)=-3$ , $q(7)=-1$ , $q(9)=0$ . Is it concave up, down, or neither?

Assumptions1. The function $q(x)$ is represented by the given table of values. . We are asked to determine if the function could be concave up, concave down, or neither.
3. We assume that the function is differentiable and continuous.

Calculate the differences.
\begin{tabular}{|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ \\ \hline1 & -10 & - \\ \hline3 & -6 &4 \\ \hline & -3 &3 \\ \hline7 & -1 &2 \\ \hline9 &0 &1 \\ \hline\end{tabular}

Calculate the second differences.
\begin{tabular}{|c|c|c|c|} \hline $x$ & $q(x)$ & $\Delta q(x)$ & $\Delta^2 q(x)$ \\ \hline1 & -10 & - & - \\ \hline3 & -6 &4 & - \\ \hline5 & -3 &3 & -1 \\ \hline & -1 &2 & -1 \\ \hline9 &0 &1 & -1 \\ \hline\end{tabular}

The second differences are all negative. This suggests that the function is concave down because a negative second derivative indicates a concave down function.
Therefore, the function $q(x)$ could be concave down.