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Consider the following statements: Statement I: The function f(x)=|x|\,sin x is differentiable at x=0. Statement II: Even if f(x) is not differentiable at x=a and g(x) is differentiable at x=a, the product f(x)g(x) may still be differentiable at x=a. Which of the following is correct?

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

Correct answer: Statement I is false, Statement II is true

Solution

Statement I is false because the function f(x)=|x|sin x is not differentiable at x=0 due to the non-smooth behavior of the absolute value function at that point. Statement II is true as it is possible for the product of a non-differentiable function and a differentiable function to be differentiable at a point, depending on the behavior of the functions involved.

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