Question

Доказать, что если сумма двух величин постоянна, то их произведение максимально тогда и только тогда когда эти величины принимают равные значения.( Это утверждение является обобщением теоремы о постоянной сумме на случай любых, а не только положительных)

Answer 1

Пусть заданы две переменные величины x и y, связанные зависимостью x+y=a, где a - некоторое постоянное число. Тогда произведение этих чисел равно x*y=x*(a-x). Рассмотрим функцию f(x)=x*(a-x). Найдем x, при котором эта функция принимает максимальное значение.
f(x)=a*x-x²
f'(x)=a-2x
Нули производной: a-2x=0 => x=a/2.
При x < a/2: f'(x) > 0 => функция возрастает
При x > a/2: f'(x) < 0 => функция убывает
Следовательно, точка x=a/2 - точка максимума функции f(x).
Соответственно, при x=a/2 y = a-a/2=a/2. Отсюда следует, что максимум произведения x*y достигается при x=y=a/2.

Answer 2

Proof that the product of two numbers is maximized when their sum is constant

To prove that if the sum of two numbers is constant, then their product is maximized when the numbers are equal, we can use algebraic reasoning.

Let's assume we have two numbers, a and b, with a constant sum, c. Mathematically, we can express this as:

a + b = c (Equation 1)

We want to find the maximum value of the product, P, of these two numbers. Mathematically, the product is given by:

P = a * b (Equation 2)

To find the maximum value of P, we can use the method of calculus. We need to find the critical points of P by taking the derivative of P with respect to a and b, and then setting the derivatives equal to zero.

Taking the derivative of P with respect to a:

dP/da = b (Equation 3)

Taking the derivative of P with respect to b:

dP/db = a (Equation 4)

Setting Equations 3 and 4 equal to zero:

b = 0 (Equation 5)

a = 0 (Equation 6)

From Equations 5 and 6, we can see that the critical points occur when either a or b is equal to zero. However, if either a or b is zero, the product P will also be zero, which is not the maximum value.

Therefore, the critical points do not give us the maximum value of P. We need to consider other possibilities.

Let's rewrite Equation 1 as:

a = c - b (Equation 7)

Substituting Equation 7 into Equation 2:

P = (c - b) * b

Expanding and simplifying:

P = c * b - b^2 (Equation 8)

To find the maximum value of P, we can take the derivative of Equation 8 with respect to b and set it equal to zero.

Taking the derivative of Equation 8 with respect to b:

dP/db = c - 2b

Setting the derivative equal to zero:

c - 2b = 0

Solving for b:

b = c/2 (Equation 9)

Substituting Equation 9 back into Equation 7:

a = c - (c/2) = c/2

Therefore, when the sum of two numbers is constant, their product is maximized when the numbers are equal to half of the constant sum.

In conclusion, we have proven that if the sum of two numbers is constant, then their product is maximized when the numbers are equal to half of the constant sum.

Note: The search results provided by You.com did not contain relevant information to directly cite for this proof. However, the proof is based on algebraic reasoning and does not require external sources.