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Consider the following statements:
Statement-1: Let g(x) be differentiable with g(1) ≠ 0 and g(-1) ≠ 0. If Rolle's theorem cannot be applied to f(x) = (x² - 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.
- Statement-1 is true, Statement-2 is true, and Statement-2 correctly explains Statement-1
- Statement-1 is true, Statement-2 is true, but Statement-2 does not correctly explain Statement-1
- Statement-1 is false, Statement-2 is true
- Statement-1 is true, Statement-2 is false
Correct answer: Statement-1 is true, Statement-2 is false
Solution
Since f(-1)=f(1)=0 already, Rolle's on f=(x^2-1)/g(x) fails only if f is not continuous/differentiable, i.e. g has a zero in (-1,1); so Statement-1 is true. Statement-2 is false because f(a)=f(b) alone is not sufficient (continuity and differentiability are also required). Hence Statement-1 true, Statement-2 false.
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