Find the critical points of $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ Announcing the arrival of Valued Associate #679: Cesar Manara Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Find all critical points of $f(x,y) = x^3 - 12xy + 8y^3$ and state maximum, minimum, or saddle points.Conceptual question: Critical PointsFind and classify the critical pointsFinding Critical Points and Local Maxima/Minima or Saddle PointHow would you find and classify ALL of the critical points of a function of 2 variables?Critical points have to be stationary?Find the critical points of the function $f(x,y)=(x^2+y^2)e^y^2-x^2$Determine and classify all critical points of function.Why are there critical points at these points?What is the nature of the critical points of $f(x,y) = 4x^2 -12xy + 9y^2$.
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Find the critical points of $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$
Announcing the arrival of Valued Associate #679: Cesar Manara
Planned maintenance scheduled April 23, 2019 at 00:00UTC (8:00pm US/Eastern)Find all critical points of $f(x,y) = x^3 - 12xy + 8y^3$ and state maximum, minimum, or saddle points.Conceptual question: Critical PointsFind and classify the critical pointsFinding Critical Points and Local Maxima/Minima or Saddle PointHow would you find and classify ALL of the critical points of a function of 2 variables?Critical points have to be stationary?Find the critical points of the function $f(x,y)=(x^2+y^2)e^y^2-x^2$Determine and classify all critical points of function.Why are there critical points at these points?What is the nature of the critical points of $f(x,y) = 4x^2 -12xy + 9y^2$.
$begingroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,7101618
asked Mar 27 at 19:34
UchuukoUchuuko
468
$endgroup$
add a comment |
$begingroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,7101618
asked Mar 27 at 19:34
UchuukoUchuuko
468
$endgroup$
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
add a comment |
$begingroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,7101618
asked Mar 27 at 19:34
UchuukoUchuuko
468
$endgroup$
I have the function $g(x,y) = 4x^3 - 12xy + 3y^2 - 18y -5.$ The only critical points that I have found for this function are $(-1, 1)$, and $(3, 9)$. But my professor insisted that there are more critical points besides these two.
Can anyone help me find them please?
multivariable-calculus partial-derivative
multivariable-calculus partial-derivative
edited Mar 27 at 19:37
thesmallprint
2,7101618
asked Mar 27 at 19:34
UchuukoUchuuko
468
edited Mar 27 at 19:37
thesmallprint
2,7101618
asked Mar 27 at 19:34
UchuukoUchuuko
468
edited Mar 27 at 19:37
thesmallprint
2,7101618
edited Mar 27 at 19:37
thesmallprint
2,7101618
edited Mar 27 at 19:37
thesmallprint
2,7101618
2,7101618
asked Mar 27 at 19:34
UchuukoUchuuko
468
asked Mar 27 at 19:34
UchuukoUchuuko
468
asked Mar 27 at 19:34
UchuukoUchuuko
468
468
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
add a comment |
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59
add a comment |
1 Answer
1
active
oldest
votes
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
$endgroup$
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
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1 Answer
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1 Answer
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$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
$endgroup$
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
$endgroup$
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
$begingroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
$endgroup$
The critical points occur where the gradient of the scalar field is zero. In this case
$nabla g(x,y)=(12x^2-12y, 6y-12x-18)=overrightarrow0$
If you solve this system of equations you'll find that the only two points are
$(-1, 1)$ and $(3, 9)$
As a consequence of the Fundamental Theorem of Algebra, these are the only two solutions.
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
answered Mar 27 at 20:01
officialnoriaofficialnoria
112
112
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
$begingroup$
The system of equations can be interpreted as the intersection of a parabola and line, which can have at most two intersection points.
$endgroup$
– amd
Mar 27 at 20:04
add a comment |
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$begingroup$
wolfram alpha gives that those two points you list are indeed the only critical points for $g$.
$endgroup$
– thesmallprint
Mar 27 at 19:42
$begingroup$
Okay, thank you. I will tell my professor.
$endgroup$
– Uchuuko
Mar 27 at 19:44
$begingroup$
Why not publish you answer so that it can be inspected closely?
$endgroup$
– NoChance
Mar 27 at 19:59