Find the linearization of the function \(f(x) = x^3 + 2x^2 - 3x + 1\) at \(x = 2\).

Step 1 ：The formula for the linearization (or the linear approximation) of a function \(f(x)\) at a point \(a\) is given by \(L(x) = f(a) + f'(a)(x-a)\).

Step 2 ：First, we find \(f(a)\) by substituting \(x = 2\) into the function: \(f(2) = 2^3 + 2(2^2) - 3(2) + 1 = 8 + 8 - 6 + 1 = 11\).

Step 3 ：Next, we find \(f'(a)\) by first finding the derivative of \(f(x)\), which is \(f'(x) = 3x^2 + 4x - 3\), and then substituting \(x = 2\) into the derivative: \(f'(2) = 3(2^2) + 4(2) - 3 = 12 + 8 - 3 = 17\).

Step 4 ：Finally, we substitute \(f(a)\) and \(f'(a)\) into the formula for the linearization: \(L(x) = 11 + 17(x - 2)\).