Question

By Newton-Raphson method, the positive root of \(x^4 - x - 10 = 0\) is:

• A. 1.871
• B. 1.868
• C. 1.856
• D. None of these

Hint:

Newton-Raphson converges quadratically; choose initial guess close to the root.

Answer 1

The Correct Option is A

The given equation is \(x^4 - x - 10 = 0\). We need to find the positive root using the Newton-Raphson method. This method is an iterative numerical technique to find successively better approximations to the roots (or zeroes) of a real-valued function.

The Newton-Raphson formula is given by: \(x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)}\)

Here, we first need to determine the function \(f(x)\) and its derivative \(f'(x)\):

• Function: \(f(x) = x^4 - x - 10\)
• Derivative: \(f'(x) = 4x^3 - 1\)

We start with an initial guess \(x_0\). Let's take \(x_0 = 2\) (since value is near to root), and apply the Newton-Raphson formula:

1. Compute \(f(x_0) = 2^4 - 2 - 10 = 4\)
2. Compute \(f'(x_0) = 4 \cdot (2^3) - 1 = 31\)
3. Next approximation: \(x_1 = 2 - \frac{4}{31} = 1.871\)

The obtained value of \(x_1\) is 1.871, which matches with one of the given options as the positive root of the equation. Checking this using the options:

• Option 1: 1.871 - Close match
• Option 2: 1.868 - Lesser
• Option 3: 1.856 - More discrepant
• Option 4: None of these - Not applicable here since 1.871 is a suitable solution.

Thus, the positive root of the equation \(x^4 - x - 10 = 0\) using the Newton-Raphson method is 1.871.