The project starts when we choose a multivariate function, in this case, by affinity we chose the
following function:
f
(
x , y
)
=x3−x2+y2−2xy +x−3y+5
To identify the critical points, we use the gradient and equalize to zero
∇f
(
x , y
)
=
(
δ f
δx
δ f
δ y
)
=0
¿
(
3x2−2x−2y+1
2y−2x−3
)
=0
By solving the equations, we get the following critical points
{
3x2−2x−2y+1=0
2y−2x−3=0
x1=1.72 x2=−0.38
y1=3.22 , y2=1.18
P1=
(
1.72 ;3.22
)
, P2(−0.38 ;1.18)
Once we obtain the critical points, we develop the Hessian matrix to know maximums, minimums
or chairs.
Hessf
(
x, y
)
=
[
∂2f
∂ x
∂2f
∂ x ∂ y
∂2f
∂ y ∂ x
∂2f
∂ y
]
=
[
6x−2−2
−2 2
]
Evaluating in the critical points
Hessf
(
1.72;3.22
)
=
[
6x−2−2
−2 2
]
det Hessf
(
1.72 ;3.22
)
=
[
6(1.72)−2−2
−2 2
]
=12.64
if D11 >0is minimums
