method of Lagrange multipliers question

Aug 2009
33
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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

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Jan 2009
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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 .


\(\displaystyle (x,y,z) = ( \frac{1}{3} , \frac{1}{3} , \frac{1}{3} )\)
 
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Jul 2007
894
298
New Orleans
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

\(\displaystyle x^2 + x^2 =4\)

\(\displaystyle 2x^2 =4\)

\(\displaystyle x^2 = 2\)

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

so your points are

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

Now just see which is you maxs and mins.

Try following my approach for the next problem.
 
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