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Question

Assertion :f(x)={x2forxcax+bforx>c
If f(x) is differentiable at x=c, then a=2c and b=c2 Reason: A continuous function is differentiable everywhere.

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
Assertion is incorrect but Reason is correct
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Solution

The correct option is C Assertion is correct but Reason is incorrect
It is given that f(x) is differentiable at x=c and every differentiable function is countinuous. So, f(x) is continuous at x=c

limxcf(x)=limxc+f(x)=f(c)
limxcx2=limxc(ax+b) [Using def. of f(x)]
c2=ac+bb=c2ac .....(i)

Now, f(x) is differentiable at x=c
LHDx=c = RHDx=c
limxcf(x)f(c)xc=limxc+f(x)f(c)xc
limxcx2c2xc=limxc(ax+b)c2xc [Using def. of f(x)]
limxc(x+c)(xc)xc=limxcax+c2acc2xc
limxc(x+c)=limxca
a=2c ......(ii)

From (i) and (ii), we get
c2=2c2+bb=c2
Hence, a=2c and b=c2

A differentiable function is continuous everywhere, however, the converse is not always true. Hence, assertion is correct but reason is incorrect.

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