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Descomponer el número 81 en dos sumandos de forma que el producto del primer sumando por el cuadrado del segundo sea máximo.

$4$ . Descomponer el número $81$ en dos sumandos de forma que el producto del primer sumando por el cuadrado del segundo sea máximo.

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Find derivatives using the quotient rule II

Full solution

Q.

4

. Descomponer el número

81

en dos sumandos de forma que el producto del primer sumando por el cuadrado del segundo sea máximo.

Define addends $x$ and $y$ : Let's define the two addends as $x$ and $y$ , where $x + y = 81$ . We need to maximize the function $f(x, y) = x \cdot y^2$ .
Express $y$ in terms of $x$ : Since $x + y = 81$ , we can express $y$ as $y = 81 - x$ . Substitute $y$ in the function to get $f(x) = x \cdot (81 - x)^2$ .
Differentiate function $f(x)$ : Now, differentiate $f(x)$ with respect to $x$ to find the critical points. $f'(x) = (81 - x)^2 - 2x(81 - x)$ .
Set derivative equal to zero: Simplify the derivative: $f'(x) = 6561 - 324x + 3x^2 - 162x + 2x^2 = 5x^2 - 486x + 6561$ .
Solve quadratic equation: Set the derivative equal to zero to find the critical points: $5x^2 - 486x + 6561 = 0$ .
Solve quadratic equation: Set the derivative equal to zero to find the critical points: $5x^2 - 486x + 6561 = 0$ . Solve the quadratic equation using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 5$ , $b = -486$ , and $c = 6561$ .
Solve quadratic equation: Set the derivative equal to zero to find the critical points: $5x^2 - 486x + 6561 = 0$ . Solve the quadratic equation using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 5$ , $b = -486$ , and $c = 6561$ . Calculate the discriminant: $(-486)^2 - 4\cdot 5\cdot 6561 = 236196 - 131220 = 104976$ .
Solve quadratic equation: Set the derivative equal to zero to find the critical points: $5x^2 - 486x + 6561 = 0$ . Solve the quadratic equation using the quadratic formula, $x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a}$ , where $a = 5$ , $b = -486$ , and $c = 6561$ . Calculate the discriminant: $(-486)^2 - 4\cdot5\cdot6561 = 236196 - 131220 = 104976$ . Calculate the roots: $x = \frac{486 \pm \sqrt{104976}}{10} = \frac{486 \pm 324}{10}$ . So, $x = 81$ or $x = 16.2$ .

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