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Let f x( ) and g x( ) are defined and differentiable for x greater or equal than x subscript 0 end subscript and f open parentheses x subscript 0 close parentheses equals g open parentheses x subscript 0 close parentheses comma f to the power of ´ left parenthesis x right parenthesis greater than straight g to the power of straight prime left parenthesis x right parenthesis text  for  end text x greater than x subscript 0 then

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  1. f left parenthesis x right parenthesis less than g left parenthesis x right parenthesis text  for some  end text x greater than x subscript 0 end subscript    
  2. f left parenthesis x right parenthesis equals g left parenthesis x right parenthesis text  for some  end text x greater than x subscript 0 end subscript    
  3. f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text  for some  end text x greater than x subscript 0 end subscript    
  4. f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text  for some  end text x less than x subscript 0 end subscript    

    Answer:The correct answer is: f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text  for some  end text x greater than x subscript 0 end subscriptϕ left parenthesis x right parenthesis equals f left parenthesis x right parenthesis minus g left parenthesis x right parenthesis text  where  end text x element of open square brackets x subscript 0 end subscript comma b close square brackets by c element of open parentheses x subscript 0 end subscript comma b close parentheses contains as member
    ϕ to the power of ´ end exponent left parenthesis c right parenthesis equals fraction numerator ϕ left parenthesis b right parenthesis minus ϕ open parentheses x subscript 0 end subscript close parentheses over denominator b minus x subscript 0 end subscript end fraction greater than 0
    therefore f left parenthesis x right parenthesis greater than g left parenthesis x right parenthesis text  for  end text x equals b

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