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Question

On which of the following intervals is the function f open parentheses x close parentheses equals 2 blank x to the power of 2 end exponent minus log invisible function application open vertical bar x close vertical bar comma x not equal to 0 increasing?

  1. left parenthesis 1 divided by 2 comma infinity right parenthesis  
  2. left parenthesis negative infinity comma negative 1 divided by 2 right parenthesis union left parenthesis 1 divided by 2 comma infinity right parenthesis  
  3. left parenthesis negative infinity comma negative 1 divided by 2 right parenthesis union left parenthesis 01 divided by 2 right parenthesis  
  4. left parenthesis negative 1 divided by 2 comma blank 0 right parenthesis union left parenthesis 1 divided by 2 comma infinity right parenthesis  

The correct answer is: left parenthesis negative 1 divided by 2 comma blank 0 right parenthesis union left parenthesis 1 divided by 2 comma infinity right parenthesis


    We have,
    f open parentheses x close parentheses equals 2 blank x to the power of 2 end exponent minus log invisible function application vertical line x vertical line

    rightwards double arrow f open parentheses x close parentheses equals open curly brackets table row cell 2 blank x to the power of 2 end exponent minus log invisible function application x comma blank x greater than 0 end cell row cell 2 blank x to the power of 2 end exponent minus log invisible function application open parentheses negative x close parentheses comma blank x less than 0 end cell end table close

    rightwards double arrow f to the power of ´ ´ end exponent open parentheses x close parentheses equals 4 blank x minus fraction numerator 1 over denominator x end fraction f o r blank a l l blank x not equal to 0

    For f left parenthesis x right parenthesis to be increasing, we must have

    f to the power of ´ ´ end exponent open parentheses x close parentheses greater than 0

    rightwards double arrow 4 blank x minus fraction numerator 1 over denominator x end fraction greater than 0

    rightwards double arrow fraction numerator 4 blank x to the power of 2 end exponent minus 1 over denominator x end fraction greater than 0


    rightwards double arrow fraction numerator left parenthesis 2 x minus 1 right parenthesis left parenthesis 2 x plus 1 right parenthesis over denominator x end fraction greater than 0

    rightwards double arrow x open parentheses 2 x minus 1 close parentheses open parentheses 2 x plus 1 close parentheses greater than 0

    rightwards double arrow x element of left parenthesis negative 1 divided by 2 comma blank 0 right parenthesis union left parenthesis 1 divided by 2 comma blank infinity right parenthesis

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