Question

Z=-10xy^2+x^2+10x+1 при x+y=1

Answer 1

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Answer 2

To find the extreme values of the function Z = -10xy^2 + x^2 + 10x + 1 subject to the constraint x + y = 1, we can use the method of Lagrange multipliers. The method of Lagrange multipliers helps us find the critical points of a function subject to a constraint.

Step 1: Set up the Lagrange function. The Lagrange function is given by L(x, y, λ) = Z - λ(x + y - 1). Here, λ is the Lagrange multiplier.

Step 2: Find the partial derivatives of the Lagrange function with respect to x, y, and λ. ∂L/∂x = -10y^2 + 2x + 10 - λ ∂L/∂y = -20xy - λ ∂L/∂λ = -(x + y - 1)

Step 3: Set the partial derivatives equal to zero and solve the system of equations. Setting each partial derivative equal to zero: -10y^2 + 2x + 10 - λ = 0 ...(i) -20xy - λ = 0 ...(ii) -(x + y - 1) = 0 ...(iii)

Step 4: Solve for x, y, and λ using equations (i), (ii), and (iii). From equation (iii), we have x + y = 1, which implies y = 1 - x. Substitute this value of y into equation (ii): -20x(1 - x) - λ = 0 -20x + 20x^2 - λ = 0 20x^2 - 20x + λ = 0 ...(iv)

Now, use equations (i) and (iv) to solve for x and λ: From equation (i): λ = 2x + 10 - 10y^2

Substitute λ into equation (iv): 20x^2 - 20x + (2x + 10 - 10y^2) = 0 20x^2 - 18x + 10 - 10y^2 = 0

Step 5: Solve for x and y. To find x, we can use the quadratic formula: x = [18 ± √(18^2 - 4 * 20 * (10 - 10y^2))] / 40 x = [18 ± √(324 - 800 + 800y^2)] / 40 x = [18 ± √(800y^2 - 476)] / 40

Now, recall that y = 1 - x: x = [18 ± √(800(1 - x)^2 - 476)] / 40 x = [18 ± √(800 - 1600x + 800x^2 - 476)] / 40 x = [18 ± √(800x^2 - 1600x + 324)] / 40

Step 6: Find the corresponding values of y. Using y = 1 - x, we can find the values of y for the values of x obtained in step 5.

Step 7: Evaluate Z at each critical point (x, y) and find the extreme values. Now that we have the critical points (x, y), we can evaluate the function Z at each of these points and find the extreme values.

However, without specific values for x and y, it's not possible to determine the exact extreme values of Z. The solution above provides the critical points (x, y) where the extreme values of Z may occur, but to find the actual values, you need to substitute the values of x and y back into the original function Z = -10xy^2 + x^2 + 10x + 1.