2 var critical point questions

In summary, the textbook states that at a local maximum, the gradient vector must be null or undefined, and a similar reasoning applies to a local minimum. However, the conversation raises questions about the possibility of the gradient vector being undefined at a minimum and how to determine the nature of a critical point when one of the partial derivatives is undefined. The conversation also discusses the limitations of using differential techniques to find extrema and suggests resorting to the definitions of optima in certain cases. Additionally, it is noted that some functions may not have local extrema at all.
  • #1
quasar987
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I have few questions about extrema of fonctions of two variables. It is written in my textbook: "At a local maximum, the gradient vector must be nul or undefined. A similar reasoning shows that the gradient must be nul at a local minimum." Actually there was no preceeding reasoning to this statement so I don't understand.

- Why couldn't the gradient vector be undefined at the min?

- If one of the partial derivative is undefined at a certain point, does it automatically means the point is a max? If no, how do you tell analytically?

- If (a,b) is a critical point because the gradient at (a,b) is 0 and if the test of the second order partial derivative fails (i.e. =0). How can I conclude analytically to the nature of the critial point?
 
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  • #2
The author has been sloppy!
- Why couldn't the gradient vector be undefined at the min?
It certainly can be!
.
 
  • #3
Ok, then I will rewrite question #2:

- If one of the partial derivatives is undefined at a certain point, how do I conclude analytically to the nature of the critical point?

- Could it be that a derivative in a direction other than the x or y-axis is undefined while it is defined in the direction of the x and y axis? In this case wouldn't the method of analysis of the partial derivatives fail to detect the critical point?
 
  • #4
Remember that the definition of optima don't involve derivatives at all -- when differential techniques fail, you often have to resort to the definitions to get your answers.
 
  • #5
Hurkyl said:
Remember that the definition of optima don't involve derivatives at all -- when differential techniques fail, you often have to resort to the definitions to get your answers.
And, I would think, some functions might be perverse enough to refuse yielding up where its extrema are, despite our best efforts..
 
  • #6
Even worse, there are functions that don't even have local extrema!

example: (here, p and q are relatively prime)


f(x) = 0 if x is irrational
f(p/q) = 1 - 1/q if q is even
f(p/q) = -1 + 1/q if q is odd
 
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1. What is a 2 var critical point question?

A 2 var critical point question is a question that involves finding the critical points of a function with two variables. A critical point is a point where the derivative of the function is equal to zero, indicating a potential maximum or minimum value.

2. How do you find the critical points of a 2 var function?

To find the critical points of a 2 var function, you must first take the partial derivatives of the function with respect to each variable. Then, set each partial derivative equal to zero and solve for the variables. The resulting values will be the critical points of the function.

3. What is the significance of critical points in a 2 var function?

Critical points are important in a 2 var function because they represent potential maximum or minimum values of the function. They can also help determine the shape of the function and identify any points of inflection.

4. How do you determine if a critical point is a maximum or minimum?

To determine if a critical point is a maximum or minimum, you can use the second derivative test. If the second derivative is positive at the critical point, then it is a minimum. If the second derivative is negative, then it is a maximum. If the second derivative is zero, the test is inconclusive and further analysis is needed.

5. Can a 2 var function have more than one critical point?

Yes, a 2 var function can have more than one critical point. In fact, a function with two variables can have an infinite number of critical points. However, not all critical points will necessarily be maximum or minimum points.

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