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General
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Question

If a b equals 2 a plus 3 b comma blank a greater than 0 comma blank b greater than 0, then the minimum value of a b is

  1. 12  
  2. 24  
  3. fraction numerator 1 over denominator 4 end fraction  
  4. None of these  

The correct answer is: 24


    Given,
    a b equals 2 a plus 3 b blank rightwards double arrow blank open parentheses a minus 3 close parentheses b equals 2 a blank rightwards double arrow blank b equals fraction numerator 2 a over denominator a minus 3 end fraction

    Now, let z equals a b equals fraction numerator 2 a to the power of 2 end exponent over denominator a minus 3 end fraction

    On differentiating w.r.t. x, we get

    fraction numerator d z over denominator s a end fraction equals fraction numerator 2 left square bracket open parentheses a minus 3 close parentheses 3 a minus a to the power of 2 end exponent right square bracket over denominator open parentheses a minus 3 close parentheses to the power of 2 end exponent end fraction equals fraction numerator 2 left square bracket a to the power of 2 end exponent minus 6 a right square bracket over denominator open parentheses a minus 3 close parentheses to the power of 2 end exponent end fraction

    For a minimum, put fraction numerator d z over denominator d a end fraction equals 0

    rightwards double arrow blank a to the power of 2 end exponent minus 6 a equals 0 blank rightwards double arrow blank a equals 0 comma blank 6

    At a equals 6 comma blank fraction numerator d to the power of 2 end exponent z over denominator d a to the power of 2 end exponent end fraction equals plus v e

    When, a equals 6 comma blank b equals 4

    therefore blank open parentheses a b close parentheses subscript m i n end subscript equals 6 cross times 4 equals 24

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