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Problem 10

# Classify the shaded value in each table as one of the following: a. a relative maximum b. a relative minimum c. a saddle point d. neither a relative extremum nor a saddle point $$\begin{array}{|r|r|r|r|r|r|r|} \hline & \mathbf{- 3} & \mathbf{- 2} & \mathbf{- 1} & \mathbf{0} & \mathbf{1} & \mathbf{2} \\ \hline \mathbf{- 3} & 100 & 101 & 100 & 97 & 92 & 85 \\ \hline \mathbf{- 2} & 99 & 100 & 99 & 96 & 91 & 84 \\ \hline \mathbf{- 1} & 98 & 99 & 98 & 95 & 90 & 83 \\ \hline \mathbf{0} & 91 & 92 & 91 & 88 & 83 & 76 \\ \hline \mathbf{1} & 72 & 73 & 72 & 69 & 64 & 57 \\ \hline \mathbf{2} & 35 & 36 & 35 & 32 & 27 & 20 \\ \hline \mathbf{3} & -26 & -25 & -26 & -29 & -34 & -41 \\ \hline \end{array}$$

Expert verified
The shaded value 88 in the table can be classified as neither a relative extremum nor a saddle point (option d).
See the step by step solution

## Step 1: Identify the neighbors

To analyze the shaded value, which is 88 in this table, we need to identify its surrounding 8 neighbors. $$\begin{array}{|c|c|c|} \hline 95 & \mathbf{90} & 83 \\ \hline 98 & \mathbf{88} & 76 \\ \hline 91 & \mathbf{83} & 69 \\ \hline \end{array}$$

## Step 2: Analyze the horizontal neighbors

Compare the shaded value (88) to its horizontal neighbors (90 and 83). In this case, 88 is less than 90 and greater than 83.

## Step 3: Analyze the vertical neighbors

Compare the shaded value (88) to its vertical neighbors (98 and 69). In this case, 88 is less than 98 and greater than 69.

## Step 4: Analyze the diagonal neighbors

Compare the shaded value (88) to its diagonal neighbors (95, 91, 76, and 83). In this case, 88 is less than 95 and 91, and greater than 76 and 83.

## Step 5: Classify the shaded value

Based on the comparison to its neighbors, the shaded value 88 is not a relative maximum (since it's not greater than all its neighbors), and it's not a relative minimum (since it's not less than all its neighbors). Additionally, it is not a saddle point as it's neither a maximum in one direction and a minimum in another direction. Therefore, the shaded value can be classified as neither a relative extremum nor a saddle point (option d).

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