Why Can't the Gradient Vector Be Undefined at a Minimum?

  • Context: Undergrad 
  • Thread starter Thread starter quasar987
  • Start date Start date
  • Tags Tags
    Critical point Point
Click For Summary

Discussion Overview

The discussion revolves around the properties of the gradient vector in relation to local extrema of functions of two variables. Participants explore the conditions under which the gradient can be undefined or zero at local minima and maxima, as well as the implications of undefined partial derivatives on the nature of critical points.

Discussion Character

  • Exploratory
  • Technical explanation
  • Debate/contested

Main Points Raised

  • One participant questions the textbook statement that the gradient must be zero or undefined at local extrema, specifically asking why the gradient cannot be undefined at a minimum.
  • Another participant asserts that the gradient vector can indeed be undefined at a minimum, challenging the textbook's assertion.
  • A participant revises their earlier question about the implications of undefined partial derivatives, seeking clarity on how to analytically determine the nature of a critical point when one of the partial derivatives is undefined.
  • There is a suggestion that a derivative might be undefined in directions other than the x or y-axis, raising concerns about the effectiveness of analyzing partial derivatives in identifying critical points.
  • One participant emphasizes that definitions of optima do not necessarily involve derivatives, suggesting that alternative methods may be required when differential techniques are insufficient.
  • Another participant notes the existence of functions that lack local extrema altogether, providing an example of such a function with distinct behavior based on the rationality of inputs.

Areas of Agreement / Disagreement

Participants express differing views on the conditions under which the gradient can be undefined at extrema, indicating a lack of consensus on this topic. The discussion remains unresolved regarding the implications of undefined partial derivatives and the nature of critical points.

Contextual Notes

Participants highlight limitations in the textbook's reasoning and the potential for functions to behave in unexpected ways, such as lacking local extrema or having undefined derivatives in certain directions.

quasar987
Science Advisor
Homework Helper
Gold Member
Messages
4,796
Reaction score
32
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?
 
Last edited:
Physics news on Phys.org
The author has been sloppy!
- Why couldn't the gradient vector be undefined at the min?
It certainly can be!
.
 
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?
 
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.
 
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..
 
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
 
Last edited:

Similar threads

Graduate How to Find Critical Points of function f(x,y,z)
  • · Replies 9 ·
Replies
9
Views
4K
Undergrad Gradient Vectors: Perpendicular to Level Curves
  • · Replies 4 ·
Replies
4
Views
3K
MHB Is f(x)=ln(x)/x Increasing or Decreasing?
  • · Replies 1 ·
Replies
1
Views
3K
Undergrad Gradient descent algorithm and optimizers
  • · Replies 5 ·
Replies
5
Views
2K
Lagrange multipliers understanding
  • · Replies 8 ·
Replies
8
Views
3K
MHB Critical Points & Extrema of Multivariable Function
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad Why Does the Gradient Point Towards the Greatest Increase?
  • · Replies 3 ·
Replies
3
Views
3K
Undergrad Geometrical interpretation of gradient
  • · Replies 7 ·
Replies
7
Views
2K
Undergrad Why does the gradient vector point straight outward from a graph?
  • · Replies 2 ·
Replies
2
Views
4K
MHB Critical Points and Global Extrema Question
  • · Replies 9 ·
Replies
9
Views
3K