To identify the local maximum point for the given function, we'll evaluate the [tex]\( f(x) \)[/tex] values at each specified point. We need to find the point where [tex]\( f(x) \)[/tex] is the highest because a local maximum is defined as a value of the function that is greater than or equal to the values of the function at surrounding points.
Let's summarize the points and their [tex]\( f(x) \)[/tex] values:
[tex]\[
\begin{array}{|c|c|c|}
\hline
\text{Point} & x & f(x) \\
\hline
S & -16.40 & -7.08 \\
\hline
T & -12.84 & -0.76 \\
\hline
U & -9.27 & -1.64 \\
\hline
V & -6.33 & 0 \\
\hline
W & -1.33 & 5.91 \\
\hline
X & 0 & 4.74 \\
\hline
Y & 1.41 & 0 \\
\hline
Z & 2.40 & -6.37 \\
\hline
\end{array}
\][/tex]
After evaluating these points, it's clear that the point with the highest [tex]\( f(x) \)[/tex] value is:
- Point W: [tex]\( (-1.33, 5.91) \)[/tex]
Therefore, the local maximum point for the function is:
[tex]\((-1.33, 5.91)\)[/tex]
Among the provided choices, the one that matches our determined local maximum is:
[tex]\[
(-\infty, 5.91)
\][/tex]
However, you might note there's a typographical error since [tex]\(-\infty\)[/tex] is not a coordinate here. The correct coordinate for maximum is:
[tex]\(( -1.33, 5.91 )\)[/tex].
Thus, the best representative local maximum point among the given correct choices is actually:
[tex]\((-1.33, 5.91)\)[/tex] (even though it is not available in the provided choices, it's the correct representation)
None of the provided choices match exactly, possibly there's an error in question typing, so correct maximum point is [tex]\((-1.33, 5.91)\)[/tex]
