ABSTRACT
Given: A scalar function V = f (y) of a single variable y.
The extrema points of a function are defined as those where its slope vanishes:
dV dy
= d dy
f (y) = 0
necessary
exist and are continous, we can expand f (y) in the Taylor series about y∗:
4V = 4 f (y) = f (y∗+4y)− f (y∗) = d f dy
∣∣∣∣∣∣ y=y∗ 4y+ 1
2 d2 f dy2
∣∣∣∣∣∣ y=y∗ 4y2 + H.O.T. (8.1)
Neglecting higher order terms
4V = 4 f (y) ≈ d f dy
∣∣∣∣∣∣ y=y∗ 4y + 1
2 d2 f dy2
∣∣∣∣∣∣ y=y∗ 4y2
The first term is called the “First Variation” δ f or the “variation” and the
second term is called the Second Variation. At the extrema point y = y∗ it is
necessary that the first variation, δ f = d f dy 4y, vanishes for an arbitrarily small
4y. Thus,
d f dy
= 0 at y = y∗ Necessary condition for an extrema
Change in f (y) in the extremal point neighborhood is approximated by the
(4y)2 term. The classification of the extremals is given by the following:
d f dy
∣∣∣∣∣∣ y=y∗
= 0 and d2 f dy2
∣∣∣∣∣∣ y=y∗
> 0, then f (y) has a local minimum
< 0, then f(y) has a local maximum
= 0, then f (y) has a “saddle” point
Example 8.1:
Find the extremal and its classification for
f (y) = tan−1 y − tan−1 ky , 0 < k < 1
d f dy
= d dy
tan−1 y − d dy
tan−1 ky = 1
1 + y2 − k
1 + (ky)2 = 0
or
(1 + (ky)2) − k(1 + y2) = 0
or
y = y∗ =
√ 1 k
Taking the second derivative at y = y∗
d2 f dy2
=
[ −2y (1 + y2)2
+ 2k3y
(1 + k2y2)2
] ∣∣∣∣∣∣ x= √
= −2k3/2(1 − k)
(1 + k)2
For the given k,
d2 f dy2
< 0 yielding a maximum at y∗ =
√ 1 k
2. Extrema of a function of several variables.
