To find the Lagrange Polynomial $f(x)$ for the given data points and calculate $f(12)$, follow these steps:

1. Lagrange Polynomial Formula:

The Lagrange polynomial is given by:

$f(x) = \sum_{i=0}^{n} y_i \cdot L_i(x)$

where $L_i(x)$ is the Lagrange basis polynomial for $x_i$:

$L_i(x) = \prod_{\substack{j=0 \\ j \neq i}}^{n} \frac{x - x_j}{x_i - x_j}$

2. Given Data Points:

The data points are: $(x_0, y_0) = (1, 18), \, (x_1, y_1) = (3, -8), \, (x_2, y_2) = (5, 30), \, (x_3, y_3) = (6, 88)$

3. Calculate $L_i(x)$:

For each $i$, compute $L_i(x)$:

For $L_0(x)$: