Step 1: Identifying the nature of the problem.
The given problem is to find the value of \( x \) for which \[ f(x) = 0 \] When the equation is nonlinear, direct algebraic methods usually fail or become very complicated.
Step 2: Classifying numerical methods by their application.
Numerical techniques are designed for different types of problems:
– Some methods solve differential equations.
– Some methods solve systems of linear equations.
– Some methods are specifically meant for finding roots of equations.
Step 3: Focusing on root-finding techniques.
A root-finding method repeatedly improves an initial guess until the function value becomes zero or nearly zero.
Among numerical methods, the Newton–Raphson technique directly targets this goal by refining guesses using the slope of the curve.
Step 4: Key working principle.
The method uses the slope (derivative) of the function to predict a better root approximation:
\[ x_{n+1} = x_n - \frac{f(x_n)}{f'(x_n)} \] Each iteration moves closer to the point where the curve crosses the \( x \)-axis.
Step 5: Eliminating incorrect options.
(A) Euler’s method → Designed for solving differential equations.
(B) Runge–Kutta method → Also used for differential equations.
(D) Gauss elimination method → Used for solving linear algebraic systems.
(C) Newton–Raphson method → Specifically developed for finding roots of nonlinear equations.
Final Conclusion:
The numerical technique most commonly used to find roots of nonlinear equations is:
\[ \boxed{\text{Newton–Raphson method}} \]