Use Lagrange multipliers to determine if the function f(x, y) = 5 + x^2 - y^2 has a
relative extrema subject to the constraint x^2 - 2y^2 = 5.
relative extrema subject to the constraint x^2 - 2y^2 = 5.
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To use Lagrange's method, we first take gradients of both the objective and constraint functions. We'll call the constraint function g.
grad f = (2x, -2y)
grad g = (2x, -4y)
Now, we want to find a constant k such that grad f = k * grad g. In other words, we want k such that
2x = 2kx and
-2y = -4ky.
Solving the first equation gives k = 1, but the second equation gives k = 1/2. So there is no k that satisfies this system of equations; therefore, Lagrange's method tells us that the objective function has no extrema when subject to the constraint.
grad f = (2x, -2y)
grad g = (2x, -4y)
Now, we want to find a constant k such that grad f = k * grad g. In other words, we want k such that
2x = 2kx and
-2y = -4ky.
Solving the first equation gives k = 1, but the second equation gives k = 1/2. So there is no k that satisfies this system of equations; therefore, Lagrange's method tells us that the objective function has no extrema when subject to the constraint.