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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?
- Statement I is false, Statement II is true
- Statement I is true, Statement II is true, and Statement II correctly explains Statement I
- Statement I is true, Statement II is false
- Statement I is true, Statement II is true, but Statement II does not explain Statement I
Correct answer: Statement I is true, Statement II is true, and Statement II correctly explains Statement I
Solution
f'(0)=lim h->0 (|h|sin h)/h = lim (sin h)(|h|/h)*... evaluating both sides gives 0, so f is differentiable at 0: Statement I is TRUE. Statement II is a true general fact (e.g. this very function), and it correctly explains why I holds. So both statements are true and II explains I.
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