# method of Lagrange multipliers question

#### quah13579

Can someone tell me how to use method of Lagrange multipliers to do following question:

Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

(a) Optimise f(x, y) = x + y,
subject to the constraint: x^2 + y^2 - 4 = 0

(b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2

#### simplependulum

MHF Hall of Honor
Can someone tell me how to use method of Lagrange multipliers to do following question:

Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

(a) Optimise f(x, y) = x + y,
subject to the constraint: x^2 + y^2 - 4 = 0

(b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2

(a)

$$\displaystyle f = x+y + a ( x^2 + y^2 - 4 )$$

we have

$$\displaystyle f_x = 1 + 2ax = 0$$ and

$$\displaystyle f_y = 1 + 2ay = 0$$

$$\displaystyle x = y = - \frac{1}{2a}$$

Sub. $$\displaystyle x=y$$ into the constraint $$\displaystyle x^2 + y^2 = 4$$

$$\displaystyle 2x^2 = 4 , x = \sqrt{2} ~ or~ -\sqrt{2}$$

$$\displaystyle (x,y) = ( \sqrt{2}, \sqrt{2}) ,( -\sqrt{2}, \sqrt{2}) ,( \sqrt{2}, -\sqrt{2}),( -\sqrt{2}, -\sqrt{2})$$

(b)

$$\displaystyle f= x^2 + y^2 + z^2 + a( x + y + z -1 ) + b( x + 2y + 3z -2)$$

$$\displaystyle f_x = 2x + a + b = 0$$ (1)
$$\displaystyle f_y = 2y + a + 2b = 0$$ (2)
$$\displaystyle f_x = 2z + a + 3b = 0$$ (3)

$$\displaystyle x = -\frac{1}{2} (a+b)$$
$$\displaystyle y = -\frac{1}{2} (a+2b)$$
$$\displaystyle z = -\frac{1}{2} (a+3b)$$

It contructs a plane , also from the two constraints , we have three planes now , solve the system of three linear equations .

$$\displaystyle (x,y,z) = ( \frac{1}{3} , \frac{1}{3} , \frac{1}{3} )$$

• quah13579

#### 11rdc11

Can someone tell me how to use method of Lagrange multipliers to do following question:

Use the method of Lagrange multipliers to find the critical points of the following constrained optimisation problems. Do not attempt to test the nature of these critical points.

(a) Optimise f(x, y) = x + y,
subject to the constraint: x^2 + y^2 - 4 = 0

(b) Optimise f(x, y, z) = x^2 + y^2 + z^2,
subject to the constraints: x + y + z = 1 and x + 2y + 3z = 2
A)

$$\displaystyle 1 = \lambda 2x$$

$$\displaystyle 1 = \lambda 2y$$

$$\displaystyle x^2 + y^2 = 4$$

$$\displaystyle x = \frac{1}{2\lambda} = y$$

so plug into your other equation

$$\displaystyle x^2 + x^2 =4$$

$$\displaystyle 2x^2 =4$$

$$\displaystyle x^2 = 2$$

$$\displaystyle x = ^+_- \sqrt{2}$$

$$\displaystyle (\sqrt{2},\sqrt{2}),(-\sqrt{2},\sqrt{2}),(-\sqrt{2},-\sqrt{2}),(\sqrt{2},-\sqrt{2})$$
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