Find and classify the critical points of the function
f(x,y)=2x^3+6xy^2−3y^3−150x
Answers
Answer:
To find the critical points of the function
f
(
x
,
y
)
, we generate and solve a system of equations in which each partial derivative is set equal to
0
. The solutions to the system are the critical points. Then, the second partial derivative test allows us to classify each critical point as a maximum, minimum or saddle point. To implement the test, we compute the value
D
(
a
,
b
)
=
f
x
x
(
a
,
b
)
f
y
y
(
a
,
b
)
−
(
f
x
y
(
a
,
b
)
)
2
for each critical point
(
a
,
b
)
using the expressions for the second partial derivatives. Once we have this information, the following guidelines allow us to classify each critical point:
1. If
D
(
a
,
b
)
>
0
and
f
x
x
(
a
,
b
)
>
0
then
(
a
,
b
)
is a minimum.
2. If
D
(
a
,
b
)
>
0
and
f
x
x
(
a
,
b
)
<
0
then
(
a
,
b
)
is a maximum.
3. If
D
(
a
,
b
)
<
0
then
(
a
,
b
)
is a saddle point.
4. If
D
(
a
,
b
)
=
0
then the test is inconclusive.
Step-by-step explanation:
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Answer:
fxy 2 0)+-$+*/$$//3