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Consider the following statements: Statement-1: Let g(x) be differentiable with g(1) \u2260 0 and g(-1) \u2260 0. If Rolle's theorem cannot be applied to f(x) = (x^2 - 1)/g(x) on the interval [-1,1], then g(x) must have at least one zero in (-1,1). Statement-2: Whenever f(a) = f(b), Rolle's theorem can be applied on the open interval (a,b). Choose the correct option.

1. Statement-1 is true, Statement-2 is true, and Statement-2 correctly explains Statement-1
2. Statement-1 is true, Statement-2 is true, but Statement-2 does not correctly explain Statement-1
3. Statement-1 is false, Statement-2 is true
4. Statement-1 is true, Statement-2 is false

Correct answer: Statement-1 is true, Statement-2 is true, but Statement-2 does not correctly explain Statement-1

Solution

Statement-1 is true because if Rolle's theorem cannot be applied, it indicates that the function g(x) must have a zero in the interval, as g(1) and g(-1) are non-zero. Statement-2 is also true, as it correctly states the condition for applying Rolle's theorem, but it does not provide a reason for the failure of the theorem in Statement-1.

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