To determine the largest interval where the function [tex]\( f(x) = -x^3 + 4x + 3 \)[/tex] is increasing, we will analyze the values given in the table. We need to compare [tex]\( f(x) \)[/tex] values for consecutive values of [tex]\( x \)[/tex] to see where [tex]\( f(x) \)[/tex] is increasing.
Here is the table again:
[tex]\[
\begin{array}{|c|c|}
\hline
x & f(x) \\
\hline
-3 & 18 \\
\hline
-2 & 3 \\
\hline
-1 & 0 \\
\hline
0 & 3 \\
\hline
1 & 6 \\
\hline
2 & 3 \\
\hline
\end{array}
\][/tex]
We proceed by comparing [tex]\( f(x) \)[/tex] values for consecutive [tex]\( x \)[/tex] values to find where [tex]\( f(x) \)[/tex] is increasing:
1. From [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-3) = 18, \quad f(-2) = 3 \quad (\text{decreasing, since } 18 > 3)
\][/tex]
2. From [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-2) = 3, \quad f(-1) = 0 \quad (\text{decreasing, since } 3 > 0)
\][/tex]
3. From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]:
[tex]\[
f(-1) = 0, \quad f(0) = 3 \quad (\text{increasing, since } 0 < 3)
\][/tex]
4. From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]:
[tex]\[
f(0) = 3, \quad f(1) = 6 \quad (\text{increasing, since } 3 < 6)
\][/tex]
5. From [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
f(1) = 6, \quad f(2) = 3 \quad (\text{decreasing, since } 6 > 3)
\][/tex]
Identifying and summarizing the intervals where the function is increasing:
- From [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]
- From [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]
Among these, both intervals are increasing. To determine the largest interval, we notice that each interval is of length 1.
Thus, the largest interval where the function [tex]\( f(x) \)[/tex] is increasing is:
[tex]\[
(-1, 0) \text{ and } (0, 1)
\][/tex]
The final answer is:
[tex]\[
\begin{aligned}
&(-1, 0), \\
&(0, 1)
\end{aligned}
\][/tex]
