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Let f and g be real valued functions def...

Let f and g be real valued functions defined on interval `(-1, 1)` such that g'' (x) is continuous, ` g(0) ne 0, g'(0) = 0, g''(0) ne 0, and f(x) = g''(0) ne 0 , and f(x) g(x) sin x`.
Statement I ` lim_( x to 0) [g(x) cos x - g(0)] [cosec x] = f''(0)`. and
Statement II f'(0) = g(0).

A

Statement I is true, Statement II is also true,
Statement II is the correct explanation of Statement I

B

Statement I is true, Statement II is also true,
Statement II is not the correct explanation of Statement I

C

Statement I is true, Statement II is false

D

Statement I is false, Statement II is true

Text Solution

Verified by Experts

The correct Answer is:
B
We have,`underset( x to 0) lim (g(x) cos x-g(0))/(sin x) " "[0/0" form "]`
`=underset( x to 0) lim (g'(x) cos x-g(x) sin x)/(cos x) = 0`
Since, `f(x) = g(x) sin x `
`f'(x) = g'(x) sin x + g(x) cos x`
and ` f''(x) = g'' (x) sin x + 2g'(x) cos x - g(x) sin x`
`rArr" " f''(0) = 0`
Thus, ` underset ( x to 0) lim [g(x) cos x - g(0) cosec x] = 0 = f''(0) `
`rArr ` Statement I is true.
Statement II` f' (x) =g'(x) sin x + g(x) cos x `
`rArr" " f'(0) = g(0) `
Statement II is not a correct explanation of Statement I.
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IIT JEE PREVIOUS YEAR-LIMIT,CONTINUITY AND DIFFERENTIABILITY -Differentiability at a point ( Assertion and Reason)
  1. Let f and g be real valued functions defined on interval (-1, 1) such ...

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