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Suppose [tex]\( f(x) \)[/tex] and [tex]\( f^{\prime}(x) \)[/tex] are continuous everywhere and have the following values:

[tex]\[
\begin{tabular}{|c|c|c|c|c|c|}
\hline
$x$ & 0 & 10 & 20 & 30 & 40 \\
\hline
$f^{\prime}(x)$ & 4 & 5 & 18 & -6 & -16 \\
\hline
\end{tabular}
\][/tex]

Based on this, determine what you can guarantee:

- Between [tex]\( x=0 \)[/tex] and [tex]\( x=10 \)[/tex], you guarantee: (Select an answer)
- Between [tex]\( x=10 \)[/tex] and [tex]\( x=20 \)[/tex], you guarantee: (Select an answer)
- Between [tex]\( x=20 \)[/tex] and [tex]\( x=30 \)[/tex], you guarantee: (Select an answer)
- Between [tex]\( x=30 \)[/tex] and [tex]\( x=40 \)[/tex], you guarantee: (Select an answer)

Sagot :

GinnyAnswer
To determine the guarantees between the given intervals, we need to look at the changes in values of the derivative [tex]\( f'(x) \)[/tex] between each pair of consecutive points. These changes represent the slopes between these intervals.

Given the values:
[tex]\[ \begin{array}{|c|c|c|c|c|c|} \hline x & 0 & 10 & 20 & 30 & 40 \\ \hline f'(x) & 4 & 5 & 18 & -6 & -16 \\ \hline \end{array} \][/tex]

Step-by-Step Solution

1. Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 10 \)[/tex]:
- The values of [tex]\( f'(x) \)[/tex] are 4 at [tex]\( x = 0 \)[/tex] and 5 at [tex]\( x = 10 \)[/tex].
- The change in [tex]\( f'(x) \)[/tex] is [tex]\( 5 - 4 = 1 \)[/tex].

Therefore, between [tex]\( x = 0 \)[/tex] and [tex]\( x = 10 \)[/tex], I guarantee a change of [tex]\( 1 \)[/tex].

2. Between [tex]\( x = 10 \)[/tex] and [tex]\( x = 20 \)[/tex]:
- The values of [tex]\( f'(x) \)[/tex] are 5 at [tex]\( x = 10 \)[/tex] and 18 at [tex]\( x = 20 \)[/tex].
- The change in [tex]\( f'(x) \)[/tex] is [tex]\( 18 - 5 = 13 \)[/tex].

Therefore, between [tex]\( x = 10 \)[/tex] and [tex]\( x = 20 \)[/tex], I guarantee a change of [tex]\( 13 \)[/tex].

3. Between [tex]\( x = 20 \)[/tex] and [tex]\( x = 30 \)[/tex]:
- The values of [tex]\( f'(x) \)[/tex] are 18 at [tex]\( x = 20 \)[/tex] and -6 at [tex]\( x = 30 \)[/tex].
- The change in [tex]\( f'(x) \)[/tex] is [tex]\( -6 - 18 = -24 \)[/tex].

Therefore, between [tex]\( x = 20 \)[/tex] and [tex]\( x = 30 \)[/tex], I guarantee a change of [tex]\( -24 \)[/tex].

4. Between [tex]\( x = 30 \)[/tex] and [tex]\( x = 40 \)[/tex]:
- The values of [tex]\( f'(x) \)[/tex] are -6 at [tex]\( x = 30 \)[/tex] and -16 at [tex]\( x = 40 \)[/tex].
- The change in [tex]\( f'(x) \)[/tex] is [tex]\( -16 - (-6) = -16 + 6 = -10 \)[/tex].

Therefore, between [tex]\( x = 30 \)[/tex] and [tex]\( x = 40 \)[/tex], I guarantee a change of [tex]\( -10 \)[/tex].

In summary:

- Between [tex]\( x = 0 \)[/tex] and [tex]\( x = 10 \)[/tex], I guarantee a change of [tex]\( 1 \)[/tex].
- Between [tex]\( x = 10 \)[/tex] and [tex]\( x = 20 \)[/tex], I guarantee a change of [tex]\( 13 \)[/tex].
- Between [tex]\( x = 20 \)[/tex] and [tex]\( x = 30 \)[/tex], I guarantee a change of [tex]\( -24 \)[/tex].
- Between [tex]\( x = 30 \)[/tex] and [tex]\( x = 40 \)[/tex], I guarantee a change of [tex]\( -10 \)[/tex].
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