Exams › JEE Main › Maths
Consider the following statements:
Statement I: If f(0)=0 and f'(x)=ln l(x+√(1+x²) r), then f(x) remains positive for every real x.
Statement II: The function f(x) increases for x>0 and decreases for x<0.
- Statement I is true, Statement II is true, and Statement II correctly explains Statement I
- Statement I is true, Statement II is true, but Statement II does not correctly explain Statement I
- Statement I is false, Statement II is true
- Statement I is true, Statement II is false
Correct answer: Statement I is true, Statement II is true, and Statement II correctly explains Statement I
Solution
f'(x)=sinh^-1(x), which is positive for x>0 and negative for x<0, so f increases for x>0 and decreases for x<0 (Statement II true). Thus f has its minimum at x=0 where f(0)=0, so f(x)>=0 for all x (Statement I true). Statement II (the monotonic behaviour) is exactly why f never goes negative, so II correctly explains I.
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