The sum of two numbers is 20. If the product | vedclass

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Question

(A) Let the two numbers be $x$ and $y$. Given that $x + y = 20$, so $y = 20 - x$. Let the product be $P = x^2 y^3$. Substituting $y$, we get $P(x) = x

Vedclass · Accepted Answer

$12, 8$

Vedclass · Answer

(A) Let the two numbers be $x$ and $y$. Given that $x + y = 20$, so $y = 20 - x$. Let the product be $P = x^2 y^3$. Substituting $y$, we get $P(x) = x^2(20 - x)^3$. To find the maximum, differentiate $P$ with respect to $x$: $\frac{dP}{dx} = 2x(20 - x)^3 + x^2 \cdot 3(20 - x)^2(-1) = x(20 - x)^2 [2(20 - x) - 3x] = x(20 - x)^2 (40 - 5x)$. Setting $\frac{dP}{dx} = 0$, we get $x = 0$, $x = 20$, or $x = 8$. Since $x$ and $y$ must be positive, we test $x = 8$. If $x = 8$, then $y = 20 - 8 = 12$. Thus, the numbers are $8$ and $12$.

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

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