Minimize f (x) = x4 − x + 1 using Newton’s method with initial point x0 = 3.

To minimize the function f(x) = x^4 - x + 1 using Newton's method, we need to follow these steps:

Step 1: Define the function and its derivative

• The function is f(x) = x^4 - x + 1
• The derivative of f(x) is f'(x) = 4x^3 - 1

Step 2: Choose an initial point x0

• In this case, the initial point is x0 = 3

Step 3: Iterate using Newton's method until convergence

• Calculate the next point x1 using the formula: x1 = x0 - f(x0) / f'(x0)
• Repeat this process until the difference between consecutive points is small enough, indicating convergence.

Let's apply these steps to find the minimum of f(x) = x^4 - x + 1 with x0 = 3:

Step 1: Define the function and its derivative

• f(x) = x^4 - x + 1
• f'(x) = 4x^3 - 1

Step 2: Choose an initial point x0

• x0 = 3

Step 3: Iterate using Newton's method until convergence

• Calculate the next point x1 using the formula: x1 = x0 - f(x0) / f'(x0)
• Repeat this process until the difference between consecutive points is small enough, indicating convergence.

Now, let's calculate the values of x1, x2, x3, and so on, until convergence is achieved.