Math  /  Calculus

QuestionCircle A or B: The one you choose will be the one graded. A. Use the method of Lagrange multipliers to find any maximum or minimal values of f(x,y,z)=3x2+y22z2f(x, y, z)=3 x^{2}+y^{2}-2 z^{2} subject to the constraint 3x+2y8z=503 x+2 y-8 z=-50.

Studdy Solution

STEP 1

What is this asking? We need to find the maximum and minimum values of a 3D function given a constraint using Lagrange multipliers. Watch out! Don't forget to check the constraint equation carefully and make sure you set up the Lagrange multiplier equation correctly!

STEP 2

1. Set up the Lagrange Multiplier Equation
2. Solve the System of Equations
3. Evaluate the Function

STEP 3

We're given the function f(x,y,z)=3x2+y22z2f(x, y, z) = 3x^2 + y^2 - 2z^2 and the constraint g(x,y,z)=3x+2y8z+50=0g(x, y, z) = 3x + 2y - 8z + 50 = 0.
Remember, we want g(x,y,z)g(x,y,z) to equal zero!

STEP 4

Now, let's **calculate the gradients**!
The gradient of ff is f=6x,2y,4z\nabla f = \langle 6x, 2y, -4z \rangle.
The gradient of gg is g=3,2,8\nabla g = \langle 3, 2, -8 \rangle.

STEP 5

The Lagrange Multiplier equation is f=λg\nabla f = \lambda \nabla g.
This gives us the equations: 6x=3λ6x = 3\lambda 2y=2λ2y = 2\lambda4z=8λ-4z = -8\lambda

STEP 6

From 6x=3λ6x = 3\lambda, we get x=12λx = \frac{1}{2}\lambda.
From 2y=2λ2y = 2\lambda, we get y=λy = \lambda.
And from 4z=8λ-4z = -8\lambda, we get z=2λz = 2\lambda.

STEP 7

Substituting these into our constraint equation 3x+2y8z+50=03x + 2y - 8z + 50 = 0, we get 3(12λ)+2(λ)8(2λ)+50=03(\frac{1}{2}\lambda) + 2(\lambda) - 8(2\lambda) + 50 = 0.

STEP 8

Simplifying, we have 32λ+2λ16λ+50=0\frac{3}{2}\lambda + 2\lambda - 16\lambda + 50 = 0, which becomes 252λ+50=0-\frac{25}{2}\lambda + 50 = 0.
Solving for λ\lambda, we find λ=4\lambda = 4.

STEP 9

Now, substitute λ=4\lambda = 4 back into our expressions for xx, yy, and zz.
We get x=12(4)=2x = \frac{1}{2}(4) = 2, y=4y = 4, and z=2(4)=8z = 2(4) = 8.

STEP 10

Finally, substitute these values into our original function f(x,y,z)=3x2+y22z2f(x, y, z) = 3x^2 + y^2 - 2z^2.
So, f(2,4,8)=3(22)+422(82)=3(4)+162(64)=12+16128=100f(2, 4, 8) = 3(2^2) + 4^2 - 2(8^2) = 3(4) + 16 - 2(64) = 12 + 16 - 128 = -100.

STEP 11

The minimum value of the function f(x,y,z)f(x, y, z) subject to the given constraint is 100-100 at the point (2,4,8)(2, 4, 8).
Since we only found one critical point using the method of Lagrange multipliers, and the function and constraint are continuous, this must be the minimum value.

Was this helpful?

Studdy solves anything!

banner

Start learning now

Download Studdy AI Tutor now. Learn with ease and get all help you need to be successful at school.

ParentsInfluencer programContactPolicyTerms
TwitterInstagramFacebookTikTokDiscord