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Maths-

General

Easy

Question

The sum of two numbers is 20. If the product of the square of one number and cube of the other is maximum, then the numbers are

12, 8
3, 4
9, 12
15, 18

The correct answer is: 12, 8

Let the numbers be $x comma blank y. comma$
$therefore blank x plus y equals 20 blank rightwards double arrow blank y equals 20 minus x$

Let $S equals x to the power of 3 end exponent open parentheses 20 minus x close parentheses to the power of 2 end exponent$

$rightwards double arrow blank fraction numerator d S over denominator d x end fraction equals 2 x to the power of 3 end exponent open parentheses 20 minus x close parentheses open parentheses negative 1 close parentheses plus 3 x to the power of 2 end exponent open parentheses 20 minus x close parentheses to the power of 2 end exponent$

For maximum or minimum, put $fraction numerator d S over denominator d x end fraction equals 0$

$rightwards double arrow blank x to the power of 2 end exponent open parentheses 20 minus x close parentheses open parentheses negative 2 x plus 60 minus 3 x close parentheses equals 0$

$rightwards double arrow blank x equals 0 comma blank x equals 12 blank$ or $x equals 20$

$x$ cannot be 0 or 20

Now $fraction numerator d S over denominator d x end fraction equals x to the power of 2 end exponent open parentheses 20 minus x close parentheses to the power of 2 end exponent left square bracket 60 minus 5 x right square bracket$

For $0 less than x less than 12 comma fraction numerator d S over denominator d x end fraction greater than 0$ and $12 less than x less than 20 comma blank fraction numerator d S over denominator d x end fraction less than 0$

$therefore blank S$ is maximum at $x equals 12$

For $x equals 12$ , then $y equals 20 minus 12 equals 8$

The required numbers are 12, 8

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