Saddle Point Requirements for f'(x)=0

  • Thread starter LagrangeEuler
  • Start date
  • Tags
    Point
In summary, it is possible for a function to have an inflection point at a particular value of x where the derivative is not equal to 0. This can be seen in the example of f(x) = x^(1/3) where the curvature changes from concave up to concave down at x = 0, despite f'(0) not being defined.
  • #1
LagrangeEuler
717
20
Is it necessary for some point ##x## of the function to be saddle that
##f'(x)=0##?
 
Last edited:
Physics news on Phys.org
  • #2
LagrangeEuler said:
Is it necessary for some point of the function to be saddle that
##f'(x)=0##?
What you have written isn't clear. For one thing, saddle points aren't applicable to functions of a single variable. For another, do you mean that f'(x) = 0 for some specific value of x? Or do you mean that f'(x) is identically equal to zero?

If you meant f'(p) = 0, for some p, all this means is that at the point (p, f(p)), the tangent to the curve is horizontal. The following three functions all have horizontal tangents when x = 0.
1. f(x) = x2 - there is a local (and global) minimum for x = 0.
2. g(x) = x3 - there is an inflection point for x = 0. This function has no minimum and no maximum.
3. h(x) = 1 - x2 - there is a local (and global) maximum for x = 0.

If you meant f'(x) ##\equiv## 0 (i.e., identically equal to zero), it must be the case that f(x) = C, the graph of which is a horizontal line. This "curve" has no minimum and no maximum.
 
  • #3
Yes I mean for particular value of ##x##. For some ##x_0## is it possible situation that
##f'(x_0)\neq 0## and that in ##x_0## function has inflection point?
 
  • #4
LagrangeEuler said:
Yes I mean for particular value of ##x##. For some ##x_0## is it possible situation that
##f'(x_0)\neq 0## and that in ##x_0## function has inflection point?
Yes. Let f(x) = x1/3.
f' is not defined at x = 0, nor is f'', but the curvature changes from concave up for x < 0 to concave down for x > 0. The fact that the curvature changes direction on either side of x = 0 is sufficient to be able to state that there is an inflection point for x = 0, which is in the domain of this function.
 

What is a saddle point?

A saddle point is a critical point on a graph of a function where the first derivative is equal to zero, but it is neither a maximum nor a minimum. Instead, it represents a change in concavity and can be thought of as a point of inflection.

How do you find saddle points?

To find saddle points, you must first take the first derivative of the function and set it equal to zero. Then, solve for the values of x that make the derivative equal to zero. These values are the potential saddle points. To confirm if they are indeed saddle points, you can use the second derivative test.

What is the second derivative test?

The second derivative test is a method used to determine if a critical point is a maximum, minimum, or saddle point. It involves taking the second derivative of the function at the critical point. If the second derivative is positive, the critical point is a minimum. If it is negative, the critical point is a maximum. If it is zero, further analysis is needed to determine if the point is a saddle point or not.

Can a function have multiple saddle points?

Yes, a function can have multiple saddle points. This occurs when the function changes concavity more than once. Each point where the first derivative is equal to zero and the second derivative is not equal to zero is a saddle point.

Why are saddle points important in optimization?

Saddle points are important in optimization because they represent points where the function is neither increasing nor decreasing. This means that the function is not at its maximum or minimum, but it could be a point of transition between the two. In optimization problems, saddle points can provide valuable information about the behavior of the function and help determine the optimal solution.

Similar threads

Replies
5
Views
1K
  • Calculus
Replies
1
Views
1K
Replies
2
Views
1K
Replies
7
Views
1K
Replies
5
Views
1K
Replies
1
Views
1K
  • Calculus
Replies
25
Views
841
Replies
11
Views
923
  • Calculus
Replies
8
Views
2K
Back
Top