InterviewSolution
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InterviewSolution
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Let the function g:`(-oo,oo)rarr (-pi //2,pi//2)` be given by g(u) `= 2 tan^(-1) (e^u)-pi/2` Then g isA. even and is strictly increasing in `(0,oo)`B. odd and is strictly decreasing `(-oo,oo)`C. odd and is strictly increasing in `(-oo,oo)`D. neither even nor odd , but is stictly increasing in `(-oo,oo)` |
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Answer» Correct Answer - C we have ` g(u)=2 tan^(-1) (e^u)-pi/2` `rArr g(u)=(2e^u)/(1+ e^(2u))gt 0 "for all u " rarr (-oo ,oo)` `rarr` g is stictly increasing function in `(-oo,oo)` Now `g(u) = tan^(-1)(e^u)-pi/2` `rArr g(u) = tan^(-1)(e^u )-(pi/2-tan ^(-1)(e^u))` `rArr g(u)=tan^(-1)(e^u)-cot^(-1)(e^(u))` `rArr g(-u)=tan ^(-1)(e^u)-cot^(-1)(e^(-u))` `rArr g(-u) = tan^(-1)(e^(1//u))-cot^(-1)(e^(1//u))` `rArr g(-u)=cot^(-1)(e^u)-tan^(-1)(e^u)=-g(u)` Hence g(u) is odd and is strictly increasing `(-oo,oo)` |
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