A root of equation f(x) = 0 can be computed to any degree of accuracy if a…
2009
A root of equation f(x) = 0 can be computed to any degree of accuracy if a 'good' initial approximation x0 is chosen for which
- A.
f (x0) > 0
- B.
f (x0) f (x0)" > 0
- C.
f (x0) f (x0)" < 0
- D.
f (x0)" > 0
Attempted by 9 students.
Show answer & explanation
Correct answer: B
For an iterative root-finding method like Newton-Raphson to converge reliably from an initial guess x0, a specific condition regarding the function's curvature must be met. The sufficient condition for convergence is that the product of the function value and its second derivative at x0 must be positive, i.e., f(x0) * f''(x0) > 0. This ensures that the tangent line at x0 does not overshoot the root excessively and guides the iteration towards the solution.