Critical points and contours

(10 pts)

In this problem, you’re going to use a couple of technological tools to help you

1. Find the critical points of a polynomial in x and y and then
2. Generate a contour plot of the function to help you classify the critical point.

To get your specific polynomial, choose your forum login name from the following list:

My polynomial in two variables is:

Modifying the Sage code, I discovered that my critical points should’ve been \left(-0.765,-0.064\right) and \left(0.445,-1.338\right).

Next, I modified the Desmos demo, and graphed my polynomial, helping me to classify my critical points:

My Polynomial in two variables is:

-1 - 2*x - x^2 - 2*y - x*y + 2*x^2*y + 2*y^2 - x*y^2 - 2*y^3

My Critical points are: (-0.86,-0.06) & (2.21,0.92)

Here’s what my graph looks like:

My polynomial function is:

1 - x + x^2 + x^3 - y + 2xy - 2y^2 - xy^2 - 2*y^3

The Critical Points:

• Min at (0.92,-1.10)

• Saddle at (0.36,-0.062)

My polynomial is:

Using the Sage code, my critical points are (-2,1) and (-1/3, -2/3).

The Desmos demo shows the following contour graph:

My function is:

1 + 2*x - 2*x^2 + 2*x^3 - y - 2*x*y - x^2*y - 2*y^2 - x*y^2 + y^3

This has a minimum at (4.27,5.78) and
a saddle point at (-0.27,1.28).

The graph looks like so:

My polynomial in two variables is f(x,y)=2x 3 −x 2 y−2x 2 −xy 2 −xy−2x−2y 3 −y 2 +y−2.

I used the Sage Code to find the critical points: (-0.353026941362916, -0.5729742173112339) and (-0.3612823674475956, 0.3587980646804176).

By using the Desmos graph below, I classified the critical points.

My polynomial is:

Using the Sage code, I found that the critical points are at about (0.60,-0.85,0.32) and (-5.69,-9.40,-101.32). Based on the contour plot, the former is a saddle point and the latter is a local minimum. The following is the Desmos contour plot:

My polynomial in two variables is:

My critical points are:

My function is:

-2 + 2*x - 2*x^2 + 2*y + 2*x*y - 2*y^2 - x*y^2 - 2*y^3

this has a minimum at (0.67,0.42),
a saddle point at (-0.17,-0.91)
and a min at (-51.8,15.5)

the graph has a leg that goes near the minimum, which can be seen in the full graph
the graph looks like so:

My function is

My critical points are: (0.274, 0.415), (-0.653, -0.303), (1.751, -0.622), (0, 0)
Here is the graph for my function:

My polynomial in two variables is:

2 - 2x - x^2 + x^3 + 2y + 2xy^2 - y^3`

My critical points are: (-0.138, -0.913) , (-0.406, 0.590) , (0.347, 1.08) , & (1.17, -0.35) and my graph looks like this:

my function is:

this has a minimum at (-1/3 , -1/3) and saddle points at (-0.2569,0.5631), (-2.5655,-1.4149), and (8.9833579, -2.37)

graph:

@jlajcin I feel like you’re missing some critical points.

My function is:
f(x,y) = 2 + x + x^2 - 2*x^3 + 2*x*y - 2*x^2*y + 2*y^2 + x*y^2
My critical points from the modified Sage Cell Server are:
[(0.617,-0.090) ,(-2.614,-15.392)]
My graph from desmos looks like:

image626×1254 79.7 KB

My polynomial is

2 - 2y + 2x^2y - y^2 + xy^2

The critical points I found were

Next, when modifying the Desmos I found this polynomial:

My polynomial in two variables is:

Modifying the Sage code, I discovered that my critical points should have been:

The desmos demo shows the following contour graph:

my polynomial is

This has a minimum at (-1.822,0.466) and a saddle point at (0,\frac{-1}{2})

The graph looks like:

My function is f(x,y) = 1 + 2x - x^2 - 2x^3 - 2xy + y^2 + y^3

This has a minimum point at (0.684,-1.086) and a saddle at (0.358,0.258)

My equation is: 2 + x + x^2 + 2y + 2xy + 2x^2y - y^2 + xy^2
The critical points in this equation are (-0.559,0.483) with a value of 2.117 and (1.67,-8.14) with a value of -38.01
the graph looks

image1920×1227 219 KB