lexipiper8296
04.01.2021 •
French
3. Le 25 décembre au matin, Elianne et sa soeur jumelle Maxim down nesoudre une en
laissée par le père Noël pour avoir le droit de deboller leur premier codcou Deux-tu le
donner un coup de main?
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Ответ:
x = 6, y =3, and the minimum = 45
Step-by-step explanation:
There are plenty of ways to solve this problem.
The Lagranian multiplier method is chosen (it is simple to apply in this case).
Given: A = x^2 + y^2
Find: x and y that satisfy 4x + 2y = 30 or 4x + 2y - 30 = 0 and A is minimum.
This is the same as finding x, y, and lambda that make the following L minimum.
L= x^2 + y^2 + lambda(4x + 2y - 30)
To find x, y, and lambda, we take the derivative of L with respect to them, set the derivative to 0, and solve the equations.
L'(x) = 2x + 4(lambda) = 0 (1)
L'(y) = 2y + 2(lambda) = 0 (2)
L'(lambda) = 4x + 2y - 30 = 0 (3)
Multiply (2) by 2 and subtract to (1), we have:
4y - 2x = 0 (*)
Combine (*) with (3) we have:
4y - 2x = 0
4x + 2y - 30 = 0
Multiply the 1st equation by 2 and add to the 2nd equation:
=> 10y - 30 = 0
=> y = 3
=> x = 4y/2 = 4(3)/2 = 6 (as (*))
=> lambda = -y = -3 (as (2))
Substitute x = 6, y = 3, and lambda = -3 into L, or substitute x = 6 and y = 3 into A to get the minimum.
Both ways would lead to the same answer.
Check:
L = x^2 + y^2 + lambda(4x + 2y - 30)
= 6^2 + 3^2 + (-3)[4(6) + 2(3) - 30]
= 36 + 9 + (-3)(0)
= 45
A = x^2 + y^2
= 36 + 9
= 45
Hope this helps!