Question

(4x+a)(4x+b)=81 имеет решение x0=a+b какое наибольшее значение принимает ab

Answer 1

Ответ:

ab = 1

Объяснение:

Подставим решение х = a + b  в уравнение (4x + a)(4x + b) = 81

(4(a + b) + a)(4(a + b) + b) = 81

(5a + 4b)(4a + 5b) = 81

20a² + 16ab + 25ab + 20b² = 81

20a² + 41ab + 20b² = 81

Известно, что

(a - b)² ≥ 0, то есть а² - 2ab + b² ≥ 0, тогда

а² + b² ≥ 2ab

Применим это неравенство и получим

20(a² + b²) ≥ 40 ab

рассмотри

20a² + 20b²  + 41ab = 81

20(a² + b²) = 81 + 41ab

81 + 41ab ≥ 40ab

81 ≥ 81ab

ab ≤ 1

Answer 2

To find the maximum value of the product ab, we need to use the given equation:

(4x + a)(4x + b) = 81

First, let's expand the left side of the equation:

(4x + a)(4x + b) = 16x^2 + 4bx + 4ax + ab

Since we know that x0 = a + b is a solution, we can substitute this value into the equation:

(4(x0) + a)(4(x0) + b) = 16(x0)^2 + 4bx0 + 4ax0 + ab (4(a + b) + a)(4(a + b) + b) = 16(a + b)^2 + 4b(a + b) + 4a(a + b) + ab (5a + 4b)(5a + b) = 16(a + b)^2 + 4b(a + b) + 4a(a + b) + ab

Now, since we know that (4x + a)(4x + b) = 81, we can also write:

(4x + a)(4x + b) - 81 = 0

Substituting the value x0 = a + b, we get:

(4(a + b) + a)(4(a + b) + b) - 81 = 0 (5a + 4b)(5a + b) - 81 = 0

Now we have two equations:

1. (5a + 4b)(5a + b) = 16(a + b)^2 + 4b(a + b) + 4a(a + b) + ab
2. (5a + 4b)(5a + b) - 81 = 0

Let's solve these equations simultaneously:

Subtract equation 2 from equation 1:

16(a + b)^2 + 4b(a + b) + 4a(a + b) + ab - 81 = 0

Now, since we are looking for the maximum value of ab, let's set the derivative of the above equation with respect to a and b equal to zero:

∂/∂a (16(a + b)^2 + 4b(a + b) + 4a(a + b) + ab - 81) = 0 ∂/∂b (16(a + b)^2 + 4b(a + b) + 4a(a + b) + ab - 81) = 0

Solving these partial derivatives, we get:

For a: 32(a + b) + 4b + 4a + b = 0 36a + 33b + 32b = 0 36a + 65b = 0 a = -(65/36) b

For b: 32(a + b) + 4b + 4a + a = 0 33a + 36b = 0 a = -(36/33) b a = -(12/11) b

Now equate the two expressions for a:

-(65/36) b = -(12/11) b

Now, since a and b cannot be equal to zero (otherwise x0 would be zero, and the equation wouldn't hold), we can divide both sides by b:

-(65/36) = -(12/11)

Now, solve for b:

65/36 = 12/11

Cross-multiply:

65 * 11 = 36 * 12

715 = 432

However, 715 is not equal to 432, which means there must be an error in the calculations or the initial assumption. After reviewing the steps, it appears that the equation (4x + a)(4x + b) = 81 does not yield a solution x0 = a + b. Therefore, the maximum value of ab cannot be determined based on the given information.