To find the local maximum of the function [tex]\( f(x) \)[/tex] using the given points, let's follow these steps:
### Step 1: Analyze the given data
First, we have data points shown in a table:
[tex]\[
\begin{tabular}{|c|c|}
\hline
$x$ & $f(x)$ \\
\hline
-4 & 16 \\
\hline
-3 & -2 \\
\hline
-2 & 0 \\
\hline
-1 & 6 \\
\hline
0 & 0 \\
\hline
1 & -2 \\
\hline
\end{tabular}
\][/tex]
### Step 2: Identify potential local maxima
Local maxima occur at points where the function value is higher than at neighboring points.
Let's examine the values closely:
- [tex]\( f(-4) = 16 \)[/tex]
- [tex]\( f(-3) = -2 \)[/tex]
- [tex]\( f(-2) = 0 \)[/tex]
- [tex]\( f(-1) = 6 \)[/tex]
- [tex]\( f(0) = 0 \)[/tex]
- [tex]\( f(1) = -2 \)[/tex]
### Step 3: Compare values
Among these values, the highest function value is [tex]\( f(-4) = 16 \)[/tex]. There are no values adjacent to -4 that are greater than 16. Therefore, the point [tex]\((-4, 16)\)[/tex] is not only the highest value but also a local maximum.
### Step 4: Conclusion
Thus, the local maximum of the function [tex]\( f(x) \)[/tex] occurs at [tex]\( x = -4 \)[/tex].
Therefore, the local maximum occurs for the [tex]\( x \)[/tex]-value [tex]\( -4 \)[/tex].
