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Continuity And Differentiability

Question
CBSEENMA12035421
Wired Faculty App

If space straight y equals fraction numerator straight a space straight x plus straight b over denominator straight c space straight x plus straight d end fraction comma space prove space that space 2 straight y subscript 1 space straight y subscript 3 equals 3 space straight y subscript 2 squared.

Solution
Here space space straight y equals fraction numerator straight a space straight x plus straight b over denominator straight c space straight x plus straight d end fraction therefore space space space space space straight y subscript 1 equals fraction numerator left parenthesis straight c space straight x plus straight d right parenthesis. straight a minus left parenthesis straight a space straight x plus straight b right parenthesis. straight c over denominator left parenthesis straight c space straight x plus straight d right parenthesis squared end fraction equals fraction numerator straight a space straight d minus straight b space straight c over denominator left parenthesis straight c space straight x plus straight d right parenthesis squared end fraction therefore space space space space space straight y to the power of 1 equals left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis squared space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 1 right parenthesis space space space space space space space space space straight y subscript 2 equals negative 2 space straight c left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 3 end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 2 right parenthesis space space space space space space space space space straight y subscript 3 equals 6 space straight c squared left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 4 end exponent space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space space... left parenthesis 3 right parenthesis straight L. straight H. straight S. equals 2 space straight y subscript 1 space straight y subscript blank cubed end subscript space space space space space space space space space equals 2 cross times left square bracket left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 2 end exponent right square bracket cross times left square bracket 6 space straight c squared left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 4 end exponent right square bracket space space space space space space space space space equals 12 space straight c squared left parenthesis straight a space straight d minus straight b space straight c right parenthesis squared left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 6 end exponent space space space space space space space space space equals 3 cross times left square bracket negative 2 space straight c left parenthesis straight a space straight d minus straight b space straight c right parenthesis left parenthesis straight c space straight x plus straight d right parenthesis to the power of negative 3 end exponent right square bracket squared equals 3 space straight y subscript 2 squared space space space space space space space space space space space space space space space space space space space space left square bracket because space of space left parenthesis 2 right parenthesis right square bracket space space space space space space space space space equals straight R. straight H. straight S.

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