Lagrange Multipliers

I.  Quiz

II.  Homework

III.  Lagrange Multipliers

Suppose that we have a function f(x,y) that we want to maximize in the restricted domain g(x,y) = c for some constant c.  A picture will be given in class.  Then we can look at the level curves of f and seek the largest level curve that intersects the curve g(x,y) = c.  It is not hard to see that these curves will be tangent.  Hence the gradient vectors will be parallel.

Theorem:  Let f(x,y) be differentiable and g(x,y) = c define a smooth curve.  Then the max and min of f subject to the constraint g occur when

f_x = lg_x  and f_y = lg_y

for some constant l.

Example:

Find the extrema of f(x,y) = x² - y² subject to the constraint y - x² = 0

Solution:  We have <2x,-2y> = lambda<-2x,1>

2x = l(-2x), -2y = l(1)

This gives us the three equations:

2x = -l2x

-2y = l (1)

y - x² = 0

the first equation gives us l= -1

Hence the second equation becomes

-2y = -1 so that y = 1/2

the third equation gives us

1/2 - x² = 0

Hence x = sqrt2/2

Example 2:

Find the distance from the origin to the surface xyz = 8

Solution:

We minimize D = x² + y² + z²  subject to the constraint

xyz = 1

We have

<2x,2y,2z> = lambda<yz,xz,xy>

2x = lambda yz

2y = lambda xz

2z = lambda xy

or

lambda = 2x/yz = 2y/xz = 2z/xy

or 2x² = 2y² = 2z²

Hence x = +- y = +- z

so that +-x³ = 8 or x = +-2

We get the points (2,2,2), (2,-2,-2), (-2,-2,2),(-2,2,-2)

these all have distance sqrt(12) from the origin.

IV.  Two constraints

Example:

Maximize xy + z  on the intersection of the two surfaces:

y² + z = 0 and x² + z = 1

Solution:

Now we set

gradf = a grad g + b grad h

<y,x,1> = a<0,2y,1> + b<2x, 0,1>

we have the five equations:

y = 2bx

x = 2ay

1 = a + b

y² + z = 0 and x² + z = 1

Using the third equation:

y = 2(1 - a)x

x = 2ay

y² + z = 0 and x² + z = 1

The first two equations give us:

a = x - y/2 = x/2y

Hence 2xy - y² = x

or x = y² /(2y - 1)

the last equations give:

z = -y² = -x² + 1

Hence -y² = -[y² /(2y - 1)]² + 1

the best way to solve this is to use the solver to get that y is about .7

Hence x = 1.2

and z = -.49