Multivariate Calculus - Bruce E. Shapiro

LECTURE 14. UNCONSTRAINED OPTIMIZATION 111
Setting ∂f/∂b = 0 gives
0 = ∂f
∂b = ∂ n∑
(mx i + b − y i ) 2
∂b
i=1
n∑
= 2 (mx i + b − y i )
= 2
i=1
n∑
(mx i + b − y i )
i=1
Dividing by 2 and separating the three sums
0 =
=
n∑
(mx i + b − y i )
i=1
n∑
mx i +
i=1
= m
Defining
n∑
b −
i=1
n∑
x i + nb −
i=1
n∑
i=1
n∑
i=1
y i
y i
X =
Y =
n∑
x i (14.8)
i=1
n∑
y i (14.9)
i=1
then we have
Next, we set ∂f/∂m = 0, which gives
0 = ∂f
∂m =
=
= 2
0 = mX + nb − Y (14.10)
∂
∂m
n∑
(mx i + b − y i ) 2
i=1
n∑
2x i (mx i + b − y i )
i=1
n∑
x i (mx i + b − y i )
i=1
Math 250, Fall 2006 Revised December 6, 2006.