Convex and Concave Functions 783(d)f= 4 x 12 + 3 x 22 + 5 x 32 + 6 x 1 x 2 +x 1 x 3 − 3 x 1 − 2 x 2 + 5: 1
H(X)=
∂^2 f/∂x 12 ∂^2 f/∂x 1 ∂x 2 ∂^2 f/∂x 1 ∂x 3
∂^2 f/∂x 1 ∂x 2 ∂^2 f/∂x 22 ∂^2 f/∂x 2 ∂x 3
∂^2 f/∂x 1 ∂x 3 ∂^2 f/∂x 2 ∂x 3 ∂^2 f/∂x 32
=
8 6 1
6 6 0
1 0 10
Here the principal minors are given by| 8 | = 8 > 0
∣
∣
∣
∣8 6
6 6
∣
∣
∣
∣=^12 >^0
∣ ∣ ∣ ∣ ∣ ∣
8 6 1
6 6 0
1 0 10
∣ ∣ ∣ ∣ ∣ ∣
= 114 > 0
and hence the matrixH(X) is positive definite for all real values ofx 1 ,x 2 , and
x 3. Therefore,f(X)is a strictly convex function.