List of top Mathematics Questions on Numerical Methods

For \(h>0\), and \(\alpha, \beta, \gamma \in \mathbb{R}\), let \[ D_h f(a) = \frac{\alpha f(a-h) + \beta f(a) + \gamma f(a+2h)}{6h} \] be a three-point formula to approximate \(f'(a)\) for any differentiable function \(f: \mathbb{R} \to \mathbb{R}\) and \(a \in \mathbb{R}\).
If \(D_h f(a) = f'(a)\) for every polynomial f of degree less than or equal to 2 and for all \(a \in \mathbb{R}\), then

Let \( f \) be a twice continuously differentiable function on \( [a, b] \) such that \( f'(x)<0 \) and \( f''(x)<0 \) for all \( x \in (a, b) \). Let \( f(\zeta) = 0 \) for some \( \zeta \in (a, b) \). The Newton-Raphson method to compute \( \zeta \) is given by \[ x_{k+1} = x_k - \frac{f(x_k)}{f'(x_k)}, \quad k = 0, 1, 2, ...... \] for an initial guess \( x_0 \). If \( x_k \in (\zeta, b) \) for some \( k \geq 0 \), then which of the following statements is/are correct?

For a fixed \(c \in \mathbb{R}\), let \(\alpha = \int_0^c (9x^2 - 5cx^4)dx\).
If the value of \(\int_0^c (9x^2 - 5cx^4)dx\) obtained by using the Trapezoidal rule is equal to \(\alpha\), then the value of c is .................. (rounded off to 2 decimal places).