Question 56

Given integrand $f(x) = x^2$.

1. Statement I (Trapezoidal Rule):

   • The global error of the composite Trapezoidal rule is given by:
     $$E_T = -\frac{(b - a)^3}{12n^2} f''(\xi), \quad \xi \in (a, b).$$
   • Here $f(x) = x^2 \implies f'(x) = 2x \implies f''(x) = 2 > 0$.
   • Because the second derivative is strictly positive, $f(x) = x^2$ is strictly convex.
   • Geometrically, the straight secant lines of trapezoids lie entirely above the convex curve $y = x^2$.
   • Consequently, the trapezoidal sum always overestimates the true area under the curve:
     $$K_{\text{trap}} \ge K_{\text{exact}}.$$
   • Therefore, Statement I is TRUE.

2. Statement II (Simpson's Rule):

   • The global error of Simpson's $1/3$ rule is given by:
     $$E_S = -\frac{(b - a)^5}{180n^4} f^{(4)}(\eta), \quad \eta \in (a, b).$$
   • For $f(x) = x^2$, the fourth derivative is identically zero ($f^{(4)}(x) = 0$).
   • Hence, the error is exactly zero: $E_S = 0$.
   • The value obtained using Simpson's rule is always strictly equal to the exact value of the definite integral.
   • Therefore, Statement II is TRUE.

Hence, Both I and II are TRUE (option C).