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 $x + y = 24$,so $y = 24 - x$ $(i)$.Let $P$ be the product of the two numbers,then $P = xy$.Substitut

Vedclass · Accepted Answer

$12, 12$

Vedclass · Answer

(A) Let the two numbers be $x$ and $y$.Given that $x + y = 24$,so $y = 24 - x$ $(i)$.Let $P$ be the product of the two numbers,then $P = xy$.Substituting $y$ from $(i)$,we get $P = x(24 - x) = 24x - x^2$.To find the maximum product,we differentiate $P$ with respect to $x$:$\frac{dP}{dx} = 24 - 2x$.For critical points,set $\frac{dP}{dx} = 0$,which gives $24 - 2x = 0$,so $x = 12$.Now,find the second derivative: $\frac{d^2P}{dx^2} = -2$.Since $\frac{d^2P}{dx^2} When $x = 12$,$y = 24 - 12 = 12$.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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