Use Lagrange multipliers to find the indicated extrema, assuming that x, and y a

Question

Use Lagrange multipliers to find the indicated extrema, assuming that x, and y are positive. Minimize f(x, y) = x^2 - 10x + y^2 - 14y + 28 Constraint: x + y = 14 Use Lagrange multipliers to find the indicated extrema of f subject to two constraints. Assume that x, y, and z are nonnegative. Maximize f(x, y, z) = xyz Constraints: x + y + z = 20, x - y + z = 4 f( ) = Consider the following. (a) Find the least squares regression line. y = (b) Calculate S, the sum of the squared errors. Use the regression capabilities of a graphing utility to verify your results.

Explanation / Answer

a) y = -0.966x + 4.441 ---> ANSWER

b) S = 4.983050847

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f(x,y) = x^2 - 10x + y^2 - 14y + 28
G(x,y) : x + y = 14
G(x,y) = x + y - 14

Fx = partial derivative with respect to x
Fx = 2x - 10

Fy = 2y - 14

Gx = 1

Gy = 1

Fx = (lambda) * Gx
2x - 10 = lamb * 1
x = (lamb + 10) / 2

Fy = lamb * Gy
2y - 14 = lamb * 1
y = (lamb + 14) / 2

x + y = 14
(lamb + 10)/2 + (lamb + 14)/2 = 14
(2lamb + 24) / 2 = 14
lamb + 12 = 14
lamb = 2

So, x = (lamb + 10)/2 --> (2 + 10)/2 ---> 6
And y = (lamb + 14)/2 --> (2 + 14) / 2 --> 8

So, f(6,8) is the minimum

f(x,y) = x^2 - 10x + y^2 - 14y + 28
So, f(6,8) = 36 - 60 + 64 - 112 + 28 = -44

So, enter it as :

f(6 , 8) = -44 ---> ANSWER

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f(x,y,z) = xyz
x + y + z = 20
x - y + z = 4

Subtracting :
(x + y + z) - (x - y + z) = 20 - 4
x + y + z - x + y - z = 16
2y = 16
y = 8

So, Maximize : 8xz
Constraint : x + z = 12
G(x,z) = x + z - 12

Fx = 8z
Fz = 8x

Gx = 1
Gz = 1

Fx = lamb * Gx
8z = lamb * 1
z = lamb/8

Fz = lamb * Gz
8x = lamb * 1
x = lamb/8

We know that x + y = 12
So, lamb/8 + lamb/8 = 12
lamb/4 = 12
So, lamb = 48

And thus, x = lamb/8 --> x = 48/8 --> x = 6
And z = lamb/8 --> z = 48/8 --> z = 6

xyz = 6 * 8 * 6 = 288

So, enter it as :

f(6 , 8 , 6) = 288 ----> ANSWER

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