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Question

If f open parentheses x close parentheses equals x e to the power of x left parenthesis x minus 1 right parenthesis end exponent, then f left parenthesis x right parenthesis is

  1. Increasing on left square bracket negative 1 divided by 21 right square bracket    
  2. Decreasing on R    
  3. Increasing on R    
  4. Decreasing on left square bracket negative 1 divided by 21 right square bracket    

The correct answer is: Increasing on left square bracket negative 1 divided by 21 right square bracket


    f open parentheses x close parentheses equals x e to the power of x left parenthesis 1 minus x right parenthesis end exponent
    rightwards double arrow f to the power of ´ end exponent open parentheses x close parentheses equals e to the power of x left parenthesis 1 minus x right parenthesis end exponent plus open parentheses 1 minus 2 x close parentheses x e to the power of x left parenthesis 1 minus x right parenthesis end exponent
    equals negative e to the power of x open parentheses 1 minus x close parentheses end exponent left parenthesis 2 x to the power of 2 end exponent minus x minus 1 right parenthesis
    equals negative e to the power of x open parentheses 1 minus x close parentheses end exponent open parentheses 2 x plus 1 close parentheses left parenthesis x minus 1 right parenthesis
    Sign scheme of f to the power of ´ end exponent left parenthesis x right parenthesis

    therefore g left parenthesis x right parenthesis is increasing in left square bracket negative 1 divided by 2 comma blank 1 right square bracket

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