How do you use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the given plane x + 8y + 5z = 24?
Now we set the respective components equal:
We can set the equations equal:
From equations 1 and 2:
From equations 1 and 3:
We can substitute these (convenient) values back into the constraint:
You may check these values:
A formal explanation (1):
Method of Lagrange Multipliers
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To use Lagrange multipliers to find the volume of the largest rectangular box in the first octant with three faces in the coordinate planes and one vertex in the plane (x + 8y + 5z = 24), follow these steps:

Define the objective function: (V = xyz), where (x), (y), and (z) are the dimensions of the rectangular box.

Formulate the constraint equation: (g(x, y, z) = x + 8y + 5z  24 = 0), representing the given plane.

Set up the Lagrangian function: (L(x, y, z, \lambda) = xyz + \lambda(x + 8y + 5z  24)), where (\lambda) is the Lagrange multiplier.

Find the partial derivatives of the Lagrangian function with respect to (x), (y), (z), and (\lambda), and set them equal to zero:
[ \begin{align*} \frac{\partial L}{\partial x} &= yz + \lambda = 0 \ \frac{\partial L}{\partial y} &= xz + 8\lambda = 0 \ \frac{\partial L}{\partial z} &= xy + 5\lambda = 0 \ \frac{\partial L}{\partial \lambda} &= x + 8y + 5z  24 = 0 \end{align*} ]

Solve this system of equations to find the critical points.

Check the critical points to determine which one yields the maximum volume.

Substitute the values of (x), (y), and (z) obtained from the critical point into the objective function (V = xyz) to find the maximum volume of the rectangular box.
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When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
When evaluating a onesided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.
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