Follow the process to find the point on the planes 3x + 3y + z = 18 and y+2z = 40 that is closest to the origin:
Use Lagrange multipliers to find the maximum value of
f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: g(x, y, z) = 3x + 3y + z = 18 and h(x, y, z) = y + 2z = 40, by following these steps.
(a) Find the coordinates of \nabla f(x, y, z)=
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>
(b) Find the coordinates of \nabla g(x, y, z)=
<
>
(c) Find the coordinates of \nabla h(x, y, z)=
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>
(d). Check the values for the lambdas:
Enter your answers as fractions (not mixed numbers).
\lambda_1 =
\lambda_2 =
(e) The maximum value of the function, subject to these constraints, satisfies the Lagrange multipliers equations. Solve for x,y, and z.
Enter your answers as fractions (not mixed numbers).
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