Classifying Extrema and Saddle Points of Multivariable Functions

In summary, the conversation discusses finding and classifying all relative extrema and saddle points of a given function. The method used involves finding the value of D, where D is equal to the product of the second derivative of x and y, minus the square of the first derivative of x and y. The resulting conclusion is that there are two points, (0,0) and (1/6,1/12), where both first derivatives are zero. This leads to the determination that (0,0) is a saddle point due to the value of D being less than 0. The conversation also mentions that the participants are likely working on the same assignment.
  • #1
Doug_West
9
0

Homework Statement



Find and classify all relative extrema and saddle points of the function
f(x; y) = xy - x^3 - y^2.

Homework Equations



D = fxx *fyy -fxy^2

The Attempt at a Solution



I got D < 0 where D = -1 and fxx = 0, when x=0 and y=0. However I am unsure as to the conclusion I should arrive at when D < 0 but fxx = 0. I'm thinking that this is a saddle point?

Thanks for the help in advance,
Dough
 
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  • #2
Doug_West said:

Homework Statement



Find and classify all relative extrema and saddle points of the function
f(x; y) = xy - x^3 - y^2.

Homework Equations



D = fxx *fyy -fxy^2

The Attempt at a Solution



I got D < 0 where D = -1 and fxx = 0, when x=0 and y=0. However I am unsure as to the conclusion I should arrive at when D < 0 but fxx = 0. I'm thinking that this is a saddle point?

Thanks for the help in advance,
Dough
You need to find locations where both of the 1st partial derivatives are zero.
 
  • #3
yep those two pts are

x=0, y=0
and
x=1/6 y=1/12
 
  • #4
evaluating pt 0,0 I get D<0 where D= -1 and fxx(0,0) = 0, so then this is a saddle point. However is it a saddle point because D<0 or because fxx = 0?
 
  • #5
guess ur doing mab127 too

when d < 0 its a saddle point doesn't matter what fxx is
 
  • #6
haha yep :D, thanks for the help.
 

1. What is a multivariable saddle point?

A multivariable saddle point is a point on a surface where there is a change in direction of the surface, known as a saddle point, in more than one direction. This means that it is a point of inflection, where the surface changes from concave to convex or vice versa. It is also known as a stationary point because the surface remains constant in all directions at that point.

2. How is a multivariable saddle point different from a single variable saddle point?

A single variable saddle point occurs on a one-dimensional curve where there is a change in direction, similar to a mountain peak or valley. A multivariable saddle point, on the other hand, occurs on a two-dimensional or higher surface where there is a change in direction in more than one direction, creating a saddle-like shape.

3. What are some real-world applications of multivariable saddle points?

Multivariable saddle points have applications in various fields such as economics, engineering, and physics. In economics, they are used to analyze the stability of equilibrium points in a market. In engineering, they are used to optimize designs for maximum stability. In physics, they are used to study the behavior of electric and magnetic fields.

4. How can multivariable saddle points be identified and calculated?

Multivariable saddle points can be identified by finding the critical points of a function, where the partial derivatives are equal to zero. These points are then evaluated using the second derivative test to determine if they are saddle points. They can also be calculated using multivariable calculus techniques such as Lagrange multipliers.

5. What is the significance of multivariable saddle points in optimization problems?

Multivariable saddle points play a crucial role in optimization problems because they represent points of maximum or minimum values on a surface. These points can be used to determine the direction of steepest ascent or descent, which is essential in finding the optimal solution for a given problem. In addition, multivariable saddle points can also be used to identify the stability of a solution in an optimization problem.

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