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Maths-

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Easy

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

12
24
$\frac{1}{4}$
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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