Sure, let's solve the equation
[tex]\[ -2x + 6 = -\left(\frac{2}{3}\right)^x + 5 \][/tex]
using three iterations of successive approximation.
### Initial Step:
First, we visually find an approximate starting point from the graph. Here, we'll use [tex]\( x = 0.75 \)[/tex] (or [tex]\( x = \frac{3}{4} \)[/tex]) as our starting point.
### Defining the Function:
Given:
[tex]\[ -2x + 6 = -\left(\frac{2}{3}\right)^x + 5 \][/tex]
Rewriting to isolate [tex]\( x \)[/tex], we define:
[tex]\[ f(x) = -2x + 6 + \left(\frac{2}{3}\right)^x - 5 \][/tex]
### First Iteration:
Starting with [tex]\( x_0 = 0.75 \)[/tex]:
[tex]\[ x_1 = f(0.75) \approx 0.2378 \][/tex]
### Second Iteration:
Using the result from the first iteration, [tex]\( x_1 = 0.2378 \)[/tex]:
[tex]\[ x_2 = f(0.2378) \approx 1.4325 \][/tex]
### Third Iteration:
Using the result from the second iteration, [tex]\( x_2 = 1.4325 \)[/tex]:
[tex]\[ x_3 = f(1.4325) \approx -1.3056 \][/tex]
### Conclusion:
Upon performing three iterations, the successive approximations yield the following sequence of values for [tex]\( x \)[/tex]:
1. Starting at [tex]\( x_0 = 0.75 \)[/tex]
2. First iteration [tex]\( x_1 \approx 0.2378 \)[/tex]
3. Second iteration [tex]\( x_2 \approx 1.4325 \)[/tex]
4. Third iteration [tex]\( x_3 \approx -1.3056 \)[/tex]
From these approximations, we see that starting at [tex]\( x = \frac{3}{4} \)[/tex] (which corresponds to choice C) as an initial guess leads to these results through successive approximations. Therefore, our starting point from the graph is validated as a good initial guess for this process.
