1. Problems with two constraints Given a differentiable function w = ƒ 1 x , y , z 2 , the goal is..

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1. Problems with two constraints Given
a differentiable function w = ƒ1x, y, z2, the
goal is to find its absolute maximum and minimum values (assuming they exist)
subject to the constraints g1x, y, z2
=
0
and h1x, y, z2
=
0,
where g
and
h
are
also differentiable.

a. Imagine a level
surface of the function ƒ and the
constraint surfaces g1x, y, z2
=
0
and h1x, y, z2
=
0.
Note that g
and
h
intersect
(in general) in a curve C on which maximum
and minimum values of ƒ must be found.
Explain why _g and _h are
orthogonal to their respective surfaces.

b. Explain why _ƒ lies
in the plane formed
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1. Problems with two constraints Given
a differentiable function w = ƒ1x, y, z2, the
goal is to find its absolute maximum and minimum values (assuming they exist)
subject to the constraints g1x, y, z2
=
0
and h1x, y, z2
=
0,
where g
and
h
are
also differentiable.

a. Imagine a level
surface of the function ƒ and the
constraint surfaces g1x, y, z2
=
0
and h1x, y, z2
=
0.
Note that g
and
h
intersect
(in general) in a curve C on which maximum
and minimum values of ƒ must be found.
Explain why _g and _h are
orthogonal to their respective surfaces.

b. Explain why _ƒ lies
in the plane formed by _g and _h at a point
of C
where
ƒ
has
a maximum or minimum value.

c. Explain why part
(b) implies that _ƒ = l_g +
m_h at a point
of C
where
ƒ
has
a maximum or minimum value, where

l and m (the
Lagrange multipliers) are real numbers.

d. Conclude from
part (c) that the equations that must be solved for maximum or minimum values
of ƒ
subject
to two

constraints are _ƒ =
l_g +
m_h, g1x, y, z2
=
0,
and h1x, y, z2
=
0.

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