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Question

The function f open parentheses x close parentheses equals fraction numerator x over denominator 2 end fraction plus fraction numerator 2 over denominator x end fraction has local minimum at

  1. x equals negative 2  
  2. x equals 0  
  3. x equals 1  
  4. x equals 2  

The correct answer is: x equals 2


    Since, f open parentheses x close parentheses equals fraction numerator x over denominator 2 end fraction plus fraction numerator 2 over denominator x end fraction
    therefore blank f to the power of ´ end exponent open parentheses x close parentheses equals fraction numerator 1 over denominator 2 end fraction minus fraction numerator 2 over denominator x to the power of 2 end exponent end fraction

    For maxima or minima, put f to the power of ´ end exponent open parentheses x close parentheses equals 0

    therefore blank fraction numerator 1 over denominator 2 end fraction minus fraction numerator 2 over denominator x to the power of 2 end exponent end fraction equals 0 blank rightwards double arrow blank x equals plus-or-minus 2

    Now, f to the power of ´ ´ end exponent open parentheses x close parentheses equals fraction numerator 4 over denominator x to the power of 3 end exponent end fraction

    rightwards double arrow blank f to the power of ´ ´ end exponent open parentheses 2 close parentheses equals fraction numerator 4 over denominator 8 end fraction equals fraction numerator 1 over denominator 2 end fraction greater than 0, minima

    And f to the power of ´ ´ end exponent open parentheses negative 2 close parentheses equals negative fraction numerator 4 over denominator 8 end fraction equals negative fraction numerator 1 over denominator 2 end fraction less than 0, maxima

    Hence, f left parenthesis x right parenthesis has local minimum at x equals 2

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