Thank you for uploading the question. Let's solve the problem using the Lagrange interpolation method to confirm the polynomial and find $y(5)$.

Given Data:

The formula for the Lagrange polynomial is:

$P(x) = \sum_{i=1}^{n} y_i \cdot \ell_i(x)$

where: $\ell_i(x) = \prod_{j=1, j \neq i}^{n} \frac{x - x_j}{x_i - x_j}$

Step 1: Compute the polynomial

We will compute the Lagrange basis polynomials $\ell_1(x), \ell_2(x), \ell_3(x), \ell_4(x)$ for $x_1 = 4, x_2 = 6, x_3 = 8, x_4 = 10$, and substitute the corresponding $y_i$ values.

1. Lagrange basis $\ell_1(x)$:

$\ell_1(x) = \frac{(x - 6)(x - 8)(x - 10)}{(4 - 6)(4 - 8)(4 - 10)} = \frac{(x - 6)(x - 8)(x - 10)}{(-2)(-4)(-6)} = -\frac{1}{48}(x - 6)(x - 8)(x - 10)$

2. Lagrange basis $\ell_2(x)$:

$\ell_2(x) = \frac{(x - 4)(x - 8)(x - 10)}{(6 - 4)(6 - 8)(6 - 10)} = \frac{(x - 4)(x - 8)(x - 10)}{(2)(-2)(-4)} = \frac{1}{16}(x - 4)(x - 8)(x - 10)$

3. Lagrange basis $\ell_3(x)$:

$\ell_3(x) = \frac{(x - 4)(x - 6)(x - 10)}{(8 - 4)(8 - 6)(8 - 10)} = \frac{(x - 4)(x - 6)(x - 10)}{(4)(2)(-2)} = -\frac{1}{16}(x - 4)(x - 6)(x - 10)$

4. Lagrange basis $\ell_4(x)$:

$\ell_4(x) = \frac{(x - 4)(x - 6)(x - 8)}{(10 - 4)(10 - 6)(10 - 8)} = \frac{(x - 4)(x - 6)(x - 8)}{(6)(4)(2)} = \frac{1}{48}(x - 4)(x - 6)(x - 8)$

Step 2: Combine the terms

The polynomial is: $P(x) = y_1 \cdot \ell_1(x) + y_2 \cdot \ell_2(x) + y_3 \cdot \ell_3(x) + y_4 \cdot \ell_4(x)$

Substitute $y_1 = 1, y_2 = 3, y_3 = 8, y_4 = 16$:

[ P(x) = 1 \cdot \left[ -\frac{1}{48}(x - 6)(x - 8)(x - 10) \right]

• 3 \cdot \left[ \frac{1}{16}(x - 4)(x - 8)(x - 10) \right] ] [
• 8 \cdot \left[ -\frac{1}{16}(x - 4)(x - 6)(x - 10) \right]
• 16 \cdot \left[ \frac{1}{48}(x - 4)(x - 6)(x - 8) \right] ]

Expanding and simplifying will give: $P(x) = \frac{3}{8}x^2 - \frac{11}{4}x + 6$

Step 3: Find $y(5)$:

Substitute $x = 5$ into $P(x)$: $P(5) = \frac{3}{8}(5^2) - \frac{11}{4}(5) + 6$ $P(5) = \frac{3}{8}(25) - \frac{55}{4} + 6 = \frac{75}{8} - \frac{110}{8} + \frac{48}{8}$ $P(5) = \frac{13}{8}$

Final Answer:

The polynomial is: $P(x) = \frac{3}{8}x^2 - \frac{11}{4}x + 6$

The value of $y(5)$ is: $y(5) = \frac{13}{8}$

Would you like further explanation or details?

Related Questions:

1. How is Lagrange interpolation used in other fields of study?
2. What are the limitations of the Lagrange interpolation method?
3. How can you verify that the polynomial passes through all given points?
4. How do higher-degree polynomials affect the accuracy of interpolation?
5. Can this method be used for data with uneven intervals in $x$?

Tip:

For large datasets, Lagrange interpolation becomes computationally intensive; consider using Newton's divided difference method for efficiency.