Welcome to Westonci.ca, your ultimate destination for finding answers to a wide range of questions from experts. Explore thousands of questions and answers from a knowledgeable community of experts ready to help you find solutions. Get detailed and accurate answers to your questions from a dedicated community of experts on our Q&A platform.
Sagot :
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]
[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]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. We appreciate your time. Please revisit us for more reliable answers to any questions you may have. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.
