Q.

If   fx=xlog cosxlog1+x2,    x≠00,                                  x=0, then

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a

f(x) is not continuous at x = 0

b

f(x) is continuous at x = 0

c

f(x) is continuous at x = 0 but not differentiable at x = 0

d

f(x) is differentiable at x = 0

answer is B.

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Detailed Solution

We  have   limx→0  fx-f0x-0=   limx→0  log cosxlog1+x2                =limx→0log1-1+cosxlog1+x21-cosx1-cosx                =limx→0log1-1-cosx1-cosx1-cosxlog1+x2                =limx→0 log1-1-cosx11-cosx2sin2x24x2x2log1+x2=-12Hence,   fx   is   differetiable  x=0.Hence, (2') and (4) are the correct answers.
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If   fx=xlog cosxlog1+x2,    x≠00,                                  x=0, then