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Question

# Assertion :Let f & g be real valued functions defined on interval (-1, 1) such that g′′(x) is continuous, g(0)≠0, g′(0)=0, gn(0)≠0 & f(x)=g(x)sinx.limx→0[g(x)cotx−g(0)cosecx]=f′′(0) Reason: f′(0)=g(0)

A
Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
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B
Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
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C
Assertion is correct but Reason is incorrect
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D
Both Assertion and Reason are incorrect
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Solution

## The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertionf(x)=g(x)sinx we have f′(x)=g(x)cosx+g′(x)sinx∴ f′(0)=g(0)cos0+g′(0)sin=g(0) f′′(x)=2g′(x)cosx−g(x)sinx+g′′(x)sinx⇒ f′′(0)=2g′(0)=0 limx→0[g(x)cotx−g(0)cosecx] =limx→0g(x)cosx−g(0)sinx=limx→0g′(x)cosx−g(x)sinxcosx (by using L-Hospital rule) g′(0)=0=f′′(0)

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