YOUR NAME: Alexander Givental
Math 53. Midterm II. October 20, 2006.
Theoretical question
(15 pts)
:
Constrained extrema and Lagrange’s method.
Let
S
be a surface in
R
3
given by the equation
g
(
x, y, z
) = 0 where
g
is a diFerentiable function,
and let
f
be another diFerentiable function in
R
3
whose domain contains
S
.
Let
f

S
denote a
function
S
→
R
obtained by restricting the domain of the function
f
to the surface
S
. By de±nition
of critical points, a point
p
∈
S
is critical for the function
f

S
, if the directional derivative of
f
at
p
in the direction of any vector
v
tangent to
S
vanishes:
D
v
f
= 0
for all
v
tangent to
S
at
p
.
Such critical points are often called
constrained extrema of the function
f
under the constraint
g
= 0
.
Theorem (Lagrange’s method).
Constrained extrema of a function
f
under a constraint
g
= 0
are in onetoone correspondence with critical points of the auxiliary function
F
(
x, y, z, λ
) :=
f
(
x, y, z
)

λg
(
x, y, z
)
.
Proof.
At a critical point
p
= (
x, y, z
) of
f

S
, for all vectors
v
tangent to
S
at
p
, we have:
0 =
D
v
f
=
∇
p
f
·
v
. Therefore the vector
∇
p
f
is perpendicular to the tangent plane to
S
at
p
and
is therefore proportional to the vector
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 Fall '08
 Lanzara
 Physics, Multivariable Calculus, Parametric equation, Conic section

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