Answer:
Explanation:
Given f(x, y) = 5x + y + 2 and g(x, y) = xy = 1
The step by step calculation and appropriate substitution is clearly shown in the attached file.
The local extreme values for the given function are;
minimum value is 2 - (2√5) while the maximum value is 2 + (2√5)
We are given the functions;
f(x, y) = 5x + y + 2 and g(x, y) = xy = 1
The general formula for lagrange multiplier is;
L(x, λ) = f(x) - λg(x)
From lagrange multipliers, we know that;
∇f = λ∇g ----(1)
Since g(x, y) = xy = 1, then;
f_x = λg_x -----(2)
f_y = λg_y -----(3)
From eq(2), we have;
λ = 5/y ------(4)
From eq 3, we have;
λ = 1/x -----(5)
Combining eq 4 and 5 gives us;
5x = y
Put 5x for y into xy = 1 to get;
5x² = 1 and so;
x = ±1/√5
Put ±1/√5 for x in xy = 1 to get;
y = ±√5
Thus, f has extreme values at;
(1/√5, √5), (-1/√5, -√5), (1/√5, -√5), (-1/√5, √5)
At (1/√5, √5), f(x, y) becomes 2 + (2√5)
At (-1/√5, -√5), f(x, y) becomes 2 - (2√5)
At (1/√5, -√5), f(x, y) becomes 2
At (-1/√5, √5), f(x, y) becomes 2
Thus, in conclusion we can say that;
The minimum value is 2 - (2√5) at the point (-1/√5, -√5) while the maximum value is 2 + (2√5) at (1/√5, √5)
Read more about Lagrange Multipliers at; https://brainly.com/question/4609414
