Finding Local Min/Max: Second Derivative Test

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In summary, When dealing with critical points of a function f(x,y), where fx and fy are the partial derivatives of z = f(x,y), we use the second derivative test to determine if the point is a local min, max, or saddle point. However, if D = 0, there is not enough information to classify the point and further analysis is needed. This is similar to the situation in single-variable calculus, where a point of inflection or a very flat extremum could occur at a point where both the first and second derivatives are equal to zero.
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eurekameh
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So we first find where fx(x,y) = 0 and fy(x,y) = 0, where fx and fy are the partial derivatives of z = f(x,y). Once we find those critical points, we use D = (fxx)(fyy) - (fxy)^2.
If D > 0 and fxx > 0, we have a local min at that point.
If D > 0 and fxx < 0, we have a local max at that point.
If D < 0, we have a saddle point.
If D = 0, no information can be found using the second derivative test.

My question is:
1. How do we deal with the D = 0 situation? How would we find if that point's a max or min?
2. What if fxx = 0?
 
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The D = 0 case has to be examined more closely, but the means for doing that are not usually discussed in a first multivariate course.

It is analogous to what happens in single-variable calculus. If both f'(a) and f''(a) equal zero, it could mean that there is a point of inflection at x = a [e.g., x = 0 for f(x) = x3 ], but there could also be a very flat maximum or minimum there [as with x = 0 for f(x) = x4 ] .

What do you check? If there is a change of concavity at x = a (that is, the sign of f''(x) changes as x passes through a ) , then there is a point of inflection there. If the concavity does not change, x = a is an extremum.

You would need to do something comparable for z = f(x,y) , but now you have to make checks in two dimensions. (Imagine the fun you can have with functions of even more variables...)[Oh, and please don't 'bump' posts in PF; it doesn't get attention any faster, it just confuses the process of having Helpers check for threads needing help. (It raises the reply count and makes it look like the thread has already been getting help...)]
 
  • #4
If [itex]f_{xx}= 0[/itex] then [itex]D\le 0[/itex].
 

Related to Finding Local Min/Max: Second Derivative Test

What is the second derivative test for finding local min/max?

The second derivative test is a method used in calculus to determine whether a critical point on a graph represents a local minimum or maximum. It involves taking the second derivative of the function at the critical point and evaluating its value.

When is the second derivative test applicable?

The second derivative test is applicable when the first derivative of the function is equal to zero at the critical point. This indicates that the slope of the function is changing from positive to negative or vice versa, which is necessary for a local min/max to exist.

How do you use the second derivative test to find local min/max?

To use the second derivative test, you must first find the critical points of the function by setting the first derivative equal to zero and solving for the variable. Then, you take the second derivative of the function at each critical point and evaluate its value. If the second derivative is positive, the critical point represents a local minimum. If it is negative, the critical point represents a local maximum.

What is the significance of the second derivative in finding local min/max?

The second derivative represents the rate of change of the slope of the function. If the second derivative is positive, the slope is increasing and the function is concave up, indicating a local minimum. If the second derivative is negative, the slope is decreasing and the function is concave down, indicating a local maximum.

Are there any limitations to using the second derivative test?

Yes, the second derivative test is only applicable to functions that are twice differentiable, meaning that their second derivative exists. It also does not work for functions with discontinuities or infinite values. In addition, the test may fail for functions with multiple critical points or inflection points.

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