Question

Consider the statements S1: If \( f(a) \cdot f(b)<0 \) then there exists a root for \( f(x) = 0 \) in between \( a \) and \( b \) S2: The Simpson's \( \frac{1}{3} \) rule approximates the definite integral \[ \int_a^b f(x)\, dx \] as sum of the areas under the parabolas. Which of the following is correct?

• A. S1 is false, S2 is true
• B. S1 is true, S2 is false
• C. S1 and S2 both are true
• D. Neither S1 nor S2 is true

Hint:

Always check for the continuity condition when applying the Intermediate Value Theorem (root existence). And remember Simpson’s \( \frac{1}{3} \) rule uses parabolic approximations.

Answer 1

The Correct Option is A

To determine the correctness of the statements, let's evaluate each one individually.

Statement S1: If \( f(a) \cdot f(b) < 0 \) then there exists a root for \( f(x) = 0 \) in between \( a \) and \( b \). This statement resembles the Intermediate Value Theorem (IVT), which states that for continuous functions on interval \([a, b]\) where \(f(a) \cdot f(b) < 0\), there is at least one root between \(a\) and \(b\). However, S1 omits the critical condition of continuity of function \(f(x)\). Without guaranteeing continuity, S1 is false. Counterexamples can be devised with discontinuous functions, for which the statement fails.

Statement S2: The Simpson's \( \frac{1}{3} \) rule approximates the definite integral \(\int_a^b f(x)\, dx\) as sum of the areas under the parabolas. Simpson's \( \frac{1}{3} \) rule is indeed a method of numerical integration that approximates the definite integral by using a parabolic approximation (second-degree polynomial) to estimate the curve \(f(x)\) between the endpoints. This statement is accurate.

Based on the evaluations: S1 is false, S2 is true.