Explanation:
(a) To find the minimum value of the function \(f(x, y) = 4x^2 + 8y^2\) subject to the constraint \(x + y = 0\), we can use the method of Lagrange multipliers.
Let's define the Lagrangian function \(L(x, y, \lambda) = f(x, y) - \lambda(g(x, y))\), where \(g(x, y) = x + y\). The parameter \(\lambda\) is the Lagrange multiplier.
Setting up the equations:
\(\frac{\partial L}{\partial x} = 0\)
\(\frac{\partial L}{\partial y} = 0\)
\(g(x, y) = 0\)
Differentiating \(L\) with respect to \(x\) and \(y\):
\(\frac{\partial L}{\partial x} = 8x - \lambda = 0\)
\(\frac{\partial L}{\partial y} = 16y - \lambda = 0\)
\(x + y = 0\)
From the first equation, we have \(8x = \lambda\), and from the second equation, we have \(16y = \lambda\). Equating these two expressions, we get \(8x = 16y\), which simplifies to \(x = 2y\).
Substituting this into the constraint equation \(x + y = 0\), we get \(2y + y = 0\), which yields \(y = 0\). From here, we can find \(x = 0\).
Therefore, the only critical point satisfying the constraint is \((x, y) = (0, 0)\).
To determine if this critical point is a minimum or maximum, we can evaluate the function \(f(x, y) = 4x^2 + 8y^2\) at this point:
\(f(0, 0) = 4(0)^2 + 8(0)^2 = 0\)
The minimum value of the function subject to the constraint \(x + y = 0\) is \(f(0, 0) = 0\).
(b) Without the specific constraint mentioned in part (b), we cannot determine the minimum value of the function. The function \(f(x, y) = 4x^2 + 8y^2\) does not have a global minimum unless there is a specific constraint or boundary condition specified.