Find the maximum and minimum of f(x; y) = 7x - 10y subject to the constraint
x2 + 4y2 = 136.
can someone tell me the final answer because my final answer has a lot of decimal point. Is it relevant?
To find the maximum and minimum of a function subject to a constraint, we can use the method of Lagrange multipliers. Here's the step-by-step process:
1. Define the function: f(x, y) = 7x - 10y
2. Define the constraint equation: x^2 + 4y^2 = 136
3. Define the Lagrangian function: L(x, y, λ) = f(x, y) - λ * g(x, y), where g(x, y) represents the constraint equation.
L(x, y, λ) = 7x - 10y - λ * (x^2 + 4y^2 - 136)
4. Find the partial derivatives of L(x, y, λ) with respect to x, y, and λ:
∂L/∂x = 7 - 2λx
∂L/∂y = -10 - 8λy
∂L/∂λ = -(x^2 + 4y^2 - 136)
5. Set the partial derivatives equal to zero and solve the resulting system of equations.
7 - 2λx = 0 => x = 7/(2λ)
-10 - 8λy = 0 => y = -10/(8λ)
x^2 + 4y^2 - 136 = 0 => (7/(2λ))^2 + 4(-10/(8λ))^2 - 136 = 0
6. Substitute the values of x and y back into the constraint equation to find the corresponding λ.
7. Once you have the value of λ, substitute it back into the equations x = 7/(2λ) and y = -10/(8λ) to find the values of x and y.
8. Finally, substitute the values of x and y into the function f(x, y) = 7x - 10y to find the maximum and minimum values.
Since this process involves solving equations, it's common to end up with decimal values. The relevance of the decimal points depends on your preference and the specific context of the problem. However, it's important to keep sufficient decimal places to maintain accuracy throughout the calculations.