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Q.

Let fx=x+loge x−x loge x, x∈0,∞Column 1 contains information about zeros of fx, f'xand .f''xColumn 2 contains information about the limiting behavior of fx, f'x and  f''xat infinity.Column 3 contains information about increasing/decreasing nature of fx and .f'xWhich of the following options is the only INCORRECT combinations?

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a

II iii P

b

I iii P

c

III i R

d

II iv Q

answer is C.

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Detailed Solution

fx=x+logx−xlgxf'x=1+2x−1−logx=1x−logxf''x=−1x2−1xf1=1>0, fe2=e2+2−2e2<0f'1=1>0, f'e=1e−1<0Letx→∞fx=−∞Letx→∞f''x=0f''x>0, for x∈0,1⇒f is increasingf'x<0, for x∈e,e2⇒f is decreasingf''x<0, for x∈0,1⇒f' is decreasingf''x<0, for x∈e,e2⇒f' is decreasing
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