To find the largest interval of [tex]\( x \)[/tex] values where the function [tex]\( f(x) = -x^3 + 4x + 3 \)[/tex] is increasing, we need to examine the given values of [tex]\( f(x) \)[/tex] provided in the table:
[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]
Next, we calculate the differences between successive [tex]\( f(x) \)[/tex] values to determine where the function is increasing.
1. Calculate differences between successive [tex]\( f(x) \)[/tex] values:
[tex]\[
\Delta f(x) = f(x_{i+1}) - f(x_i)
\][/tex]
- For [tex]\( x = -3 \)[/tex] to [tex]\( x = -2 \)[/tex]:
[tex]\[
f(-2) - f(-3) = 3 - 18 = -15
\][/tex]
- For [tex]\( x = -2 \)[/tex] to [tex]\( x = -1 \)[/tex]:
[tex]\[
f(-1) - f(-2) = 0 - 3 = -3
\][/tex]
- For [tex]\( x = -1 \)[/tex] to [tex]\( x = 0 \)[/tex]:
[tex]\[
f(0) - f(-1) = 3 - 0 = 3
\][/tex]
- For [tex]\( x = 0 \)[/tex] to [tex]\( x = 1 \)[/tex]:
[tex]\[
f(1) - f(0) = 6 - 3 = 3
\][/tex]
- For [tex]\( x = 1 \)[/tex] to [tex]\( x = 2 \)[/tex]:
[tex]\[
f(2) - f(1) = 3 - 6 = -3
\][/tex]
2. Identify the intervals where the differences are positive, indicating the function is increasing:
[tex]\[
\begin{aligned}
& \text{Interval from } x = -3 \text{ to } x = -2: \Delta f(x) = -15 \quad (\text{decreasing}) \\
& \text{Interval from } x = -2 \text{ to } x = -1: \Delta f(x) = -3 \quad (\text{decreasing}) \\
& \text{Interval from } x = -1 \text{ to } x = 0: \Delta f(x) = 3 \quad (\text{increasing}) \\
& \text{Interval from } x = 0 \text{ to } x = 1: \Delta f(x) = 3 \quad (\text{increasing}) \\
& \text{Interval from } x = 1 \text{ to } x = 2: \Delta f(x) = -3 \quad (\text{decreasing}) \\
\end{aligned}
\][/tex]
From these differences, the intervals where the function is increasing are:
[tex]\[
(-1, 0) \text{ and } (0, 1)
\][/tex]
Thus, combining these, we conclude the largest interval where the function is increasing is:
[tex]\[
(-1, 1)
\][/tex]
So, the largest interval of [tex]\( x \)[/tex] values where the function [tex]\( f(x) \)[/tex] is increasing is:
[tex]\[
\boxed{(-1, 1)}
\][/tex]
