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text  If   end text f left parenthesis x right parenthesis equals open curly brackets table attributes columnalign center center columnspacing 1em end attributes row cell open parentheses e to the power of x minus 1 close parentheses vertical line x sin invisible function application x vertical line end cell cell x greater than 0 end cell row 0 cell x to the power of n comma end cell end table space of 1em x equals 0 close is derivable at x equals 0  then interval of possible values of ‘n’ is

  1. left square bracket 1 comma midline horizontal ellipsis infinity right parenthesis
  2. left parenthesis negative infinity comma 1 right square bracket
  3. left parenthesis negative infinity comma 2 right square bracket
  4. left square bracket 1 comma 2 right square bracket

The correct answer is: left parenthesis negative infinity comma 1 right square bracket


    table attributes columnalign right columnspacing 1em end attributes row cell f to the power of straight prime left parenthesis 0 right parenthesis equals lim for x not stretchy rightwards arrow 0 of   fraction numerator fraction numerator open parentheses e to the power of x minus 1 close parentheses left parenthesis x sin invisible function application x right parenthesis over denominator x to the power of n end fraction minus 0 over denominator x end fraction equals lim for x not stretchy rightwards arrow 0 of   open parentheses fraction numerator sin invisible function application x over denominator x to the power of n minus 1 end exponent end fraction close parentheses end cell row cell equals lim for x not stretchy rightwards arrow 0 of   open parentheses 1 over x to the power of n minus 2 end exponent close parentheses not stretchy rightwards double arrow n minus 2 less or equal than 0 not stretchy rightwards double arrow n end cell row cell less or equal than 2 end cell end table

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