Q2. Find all the local extrema and saddle points, if any for the function f(x, y) =
2(In x + In y) - 4x – y
Answers
Answered by
1
Answer:
find out the local extrema and saddle points
Answered by
1
Step-by-step explanation:
The function is
f
(
x
,
y
)
=
x
2
+
x
y
+
y
2
+
3
x
−
3
y
+
4
The partial derivatives are
∂
f
∂
x
=
2
x
+
y
+
3
∂
f
∂
y
=
2
y
+
x
−
3
Let
∂
f
∂
x
=
0
and
∂
f
∂
y
=
0
Then,
{
2
x
+
y
+
3
=
0
2
y
+
x
−
3
=
0
⇒
,
{
x
=
−
3
y
=
3
∂
2
f
∂
x
2
=
2
∂
2
f
∂
y
2
=
2
∂
2
f
∂
x
∂
y
=
1
∂
2
f
∂
y
∂
x
=
1
The Hessian matrix is
H
f
(
x
,
y
)
=
⎛
⎜
⎝
∂
2
f
∂
x
2
∂
2
f
∂
x
∂
y
∂
2
f
∂
y
∂
x
∂
2
f
∂
y
2
⎞
⎟
⎠
The determinant is
D
(
x
,
y
)
=
det
(
H
(
x
,
y
)
)
=
∣
∣
∣
2
1
1
2
∣
∣
∣
=
4
−
1
=
3
>
0
Therefore,
There are no saddle points.
D
(
1
,
1
)
>
0
and
∂
2
f
∂
x
2
>
0
, there is a local minimum at
(
−
3
,
3
)
I hope you will get your answer please mark as brainlist
Similar questions