To solve the equation [tex]\(4^{-x} + 5 = 3^x + 4\)[/tex] using successive approximation, we start with an initial guess and iterate the process several times to converge to an approximate solution. We'll then select the closest approximation from the given options.
### Step-by-Step Solution
1. Equation Setup:
The given equation is:
[tex]\[
4^{-x} + 5 = 3^x + 4
\][/tex]
2. Initial Guess:
We start with an initial guess for [tex]\(x\)[/tex]. For this problem, let's use:
[tex]\[
x_0 = \frac{1}{4}
\][/tex]
3. Define the Function:
Translate the equation into a form that a function can be iteratively evaluated. Rearrange the original equation to isolate [tex]\( f(x) \)[/tex]:
[tex]\[
f(x) = 4^{-x} + 5 - 3^x - 4
\][/tex]
Simplifying, we get:
[tex]\[
f(x) = 4^{-x} + 1 - 3^x
\][/tex]
4. Iteration:
We will perform three iterations of successive approximation:
- Iteration 1:
[tex]\[
x_1 = f(x_0) = 4^{-\frac{1}{4}} + 1 - 3^{\frac{1}{4}}
\][/tex]
Calculate [tex]\(x_1\)[/tex].
- Iteration 2:
[tex]\[
x_2 = f(x_1)
\][/tex]
Use the result of [tex]\( x_1 \)[/tex] to find [tex]\( x_2 \)[/tex].
- Iteration 3:
[tex]\[
x_3 = f(x_2)
\][/tex]
Use the result of [tex]\( x_2 \)[/tex] to find [tex]\( x_3 \)[/tex].
5. Choose the Closest Approximation:
After performing these iterations, we'll compare the results with the given options to determine the closest approximate value for [tex]\(x\)[/tex]:
- A. [tex]\( x \approx \frac{1}{4} \)[/tex]
- B. [tex]\( x \approx \frac{5}{16} \)[/tex]
- C. [tex]\( x \approx \frac{3}{8} \)[/tex]
- D. [tex]\( x \approx \frac{7}{16} \)[/tex]
From our iterative process, we find that the final approximation [tex]\( x_3 \)[/tex] is closest to the value:
[tex]\[
\boxed{\frac{7}{16}}
\][/tex]
Hence, the best approximation for the solution to the equation [tex]\( 4^{-x} + 5 = 3^x + 4 \)[/tex] is [tex]\( x \approx \frac{7}{16} \)[/tex], which corresponds to option D.
