Find two numbers whose sum is 24 and whose pr | vedclass

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Question

(A) Let the two numbers be $x$ and $y$. Given that their sum is $x + y = 24$,we can express $y$ as $y = 24 - x$.Let $P$ be the product of the two numb

Vedclass · Accepted Answer

$12, 12$

Vedclass · Answer

(A) Let the two numbers be $x$ and $y$. Given that their sum is $x + y = 24$,we can express $y$ as $y = 24 - x$.Let $P$ be the product of the two numbers,so $P = x \cdot y = x(24 - x) = 24x - x^2$.To find the maximum product,we find the derivative of $P$ with respect to $x$:$\frac{dP}{dx} = 24 - 2x$.Setting the derivative to zero for critical points:$24 - 2x = 0 \Rightarrow x = 12$.To confirm this is a maximum,we find the second derivative:$\frac{d^2P}{dx^2} = -2$.Since the second derivative is negative $(-2 If $x = 12$,then $y = 24 - 12 = 12$.Thus,the two numbers are $12$ and $12$.

Find two numbers whose sum is $24$ and whose product is as large as possible.

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