Explications étape par étape:
To find the critical values for minimizing the cost function \(C = 8x^2 - xy + 12y^2\) subject to the constraint \(x + y = 42\), we can use the Lagrange multiplier method.
The Lagrangian function is given by:
\[ L(x, y, \lambda) = 8x^2 - xy + 12y^2 + \lambda(42 - x - y) \]
Now, we need to find the partial derivatives of \(L\) with respect to \(x\), \(y\), and \(\lambda\) and set them equal to zero:
\[ \frac{\partial L}{\partial x} = 16x - y - \lambda = 0 \]
\[ \frac{\partial L}{\partial y} = -x + 24y - \lambda = 0 \]
\[ \frac{\partial L}{\partial \lambda} = 42 - x - y = 0 \]
Solving this system of equations will give us the critical values. Let's solve for \(x\), \(y\), and \(\lambda\):
1. From the first equation: \(16x - y - \lambda = 0\), we can express \(y\) in terms of \(x\): \(y = 16x - \lambda\).
2. Substitute this expression for \(y\) into the second equation: \(-x + 24y - \lambda = 0\).
3. Substitute the constraint into the third equation: \(42 - x - y = 0\).
Solving this system of equations will yield the critical values for minimizing the cost function subject to the given constraint.
