Question

X*y=64. Найти такие числа чтобы их сумма была max

Answer 1

Если это x*y, значит 8*8 не подходит сразу.

64*1=64, 64+1=65;

32*2=64, 32+2=34;

16*4=64, 16+4=20;

Answer 2

чтобы найти наибольшее значение выражения х+у, составим функцию f(x)=x+y

и исследуем ее на экстремумы :

$x*y=64 \\ y=\frac{64}{x} \\ \\ f(x)=x+y \\ f(x)=x+\frac{64}{x} \\ \\ f'(x)=1-\frac{64}{x^2} =0\\ \\ \frac{x^2-64}{x^2} =0 \\ \\ \frac{(x-8)(x+8)}{x^2} =0$

далее точки экстремума находятся методом интервалов:

Корни числителя: 8; -8

Корни знаменателя: 0; 0 (в знаменателе стоит x², значит здесь два РАВНЫХ корня)

$+++[-8]----(0)----[8]+++>_x$

-8 - точка максимума

8- точка минимума

наибольшее значение выражения при x<0

$x+y=x+\frac{64}{x} =-8+\frac{64}{-8=-16}$

Если x>0, то максимального значение нет (максимальное значение x+y стремится к бесконечности)

Answer 3

To find two numbers, x and y, such that their product is 64 and their sum is maximum, we can use a bit of algebra and optimization techniques.

Let's use the AM-GM (Arithmetic Mean-Geometric Mean) inequality to solve this problem. The AM-GM inequality states that for any two positive numbers a and b:

(a + b) / 2 ≥ √(a * b)

Equality holds when a = b.

In our case, x and y are positive numbers (since they are non-zero numbers and their product is positive), so we can apply the AM-GM inequality to find the maximum sum.

Given the constraint: x * y = 64

Let's find the sum (x + y) that maximizes the value:

(x + y) / 2 ≥ √(x * y)

Substitute the value of the product:

(x + y) / 2 ≥ √(64)

(x + y) / 2 ≥ 8

Now, to maximize the sum (x + y), we want the left side to be as close to the right side as possible. To achieve this, we set x and y to be equal:

x = y

Now, from the equation x * y = 64, we can substitute y with x:

x * x = 64

x^2 = 64

Taking the square root of both sides:

x = ±√64

x = ±8

Since x and y are positive numbers, we take the positive square root:

x = 8

Now, we can find y:

y = 64 / x

y = 64 / 8

y = 8

So, the two numbers that satisfy the given condition and have the maximum sum are x = 8 and y = 8. Their sum is:

Sum (x + y) = 8 + 8 = 16

Therefore, the maximum sum of two numbers whose product is 64 is 16, and those two numbers are 8 and 8.