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Consider the following statements: Statement I: If f(0)=0 and f'(x)=\ln\bigl(x+\sqrt{1+x^2}\bigr), then f(x) remains positive for every real x. Statement II: The function f(x) increases for x>0 and decreases for x<0.

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

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

Solution

Statement I is false because the function f(x) does not remain positive for every real x; it can take negative values for certain inputs. Statement II is true as the derivative f'(x) indicates that the function increases for x > 0 and decreases for x < 0.

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