Is continuity of the second derivative necessary for the second derivative test?

In summary, the second derivative test for finding minima and maxima requires that ##f''## is defined at ##c##, but it does not have to be continuous in a region around ##c##. The first link discusses the use of the second derivative test, while the second link provides a counterexample to the assumption that ##f''## must be continuous in a region around ##c##. The author of the first link assumes continuity of ##f''## around ##c##, but this is not necessary for the proof.
  • #1
PFuser1232
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According to this link: http://tutorial.math.lamar.edu/Classes/CalcI/ShapeofGraphPtII.aspx
The second derivative test can only be applied if ##f''## is continuous in a region around ##c##.
But according to this link: http://calculus.subwiki.org/wiki/Second_derivative_test#Requirement_of_twice_differentiability
##f''## need not be defined in a region around ##c##.
I'm confused as to what is required for the second derivative test.
##f''## is allowed to have a discontinuity at ##c##, but not around ##c##? If ##f''## is continuous around ##c##, then ##\lim_{x→a} f''(x)## must equal ##f''(a)## where ##a## is in some region around ##c##, but according to the second link, ##f''(a)## doesn't have to be defined.
 
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  • #2
There is no need for ## f''## to be defined at points other than c. If ## f'(c)=0 ## and ## f''(c)>0 ##, using the definition ## f''(c)=\lim_{h\rightarrow 0}\frac{f'(c+h)-f'(c)}{h} ## this implies that within some neighborhood of ## c,f' ## is positive to the right of of ## c ## and negative to its left, so f must have a minimum at ## c ## .
 
  • #3
wabbit said:
There is no need for ## f''## to be defined at points other than c. If ## f'(c)=0 ## and ## f''(c)>0 ##, using ## f''(c)=\lim_{h\rightarrow 0}\frac{f'(x+h)}{h} ## this implies that within some neighborhood of ## c,f' ## is positive to the right of of ## c ## and negative to its left, so f must have a minimum at ## c ## .

http://tutorial.math.lamar.edu/Classes/CalcI/DerivativeAppsProofs.aspx#Extras_DerAppPf_SDT

The author assumes that ##f''## is continuous around ##c## to prove the second derivative test.
 
  • #4
MohammedRady97 said:
I'm confused as to what is required for the second derivative test.
Only that ##f''## is defined at ##c##. I don't see where in the first link it suggests a requirement for continuity: if it does it is wrong.
 
  • #5
I just gave you a proof that does not require that assumption. Did you read it ? If you don't understand it I can help you with fleshing out the details.

Whatever the author assumes is irrelevant to the question you asked about which assumption is necessary.
 
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What is the Second Derivative Test?

The Second Derivative Test is a mathematical tool used to determine the nature of a critical point on a function, such as a maximum, minimum, or saddle point. It involves taking the second derivative of the function at the critical point and analyzing its value to make a conclusion about the nature of the critical point.

How do you use the Second Derivative Test?

To use the Second Derivative Test, you must first find the critical points of the function by setting the first derivative equal to zero. Then, take the second derivative of the function and plug in the critical points. If the second derivative is positive, the critical point is a minimum. If the second derivative is negative, the critical point is a maximum. If the second derivative is zero, the test is inconclusive and further analysis is needed.

What is the difference between the First and Second Derivative Tests?

The First Derivative Test is used to determine the nature of a critical point by analyzing the sign of the first derivative at that point. It is a less accurate method compared to the Second Derivative Test, which involves analyzing the value of the second derivative at the critical point. The Second Derivative Test provides more information about the nature of the critical point, as it takes into account the curvature of the function.

Can the Second Derivative Test be used for all functions?

The Second Derivative Test can be used for all functions that are twice differentiable, meaning that their first and second derivatives exist and are continuous. However, it may not always provide a conclusive result, as there are cases where the second derivative is zero at a critical point, requiring further analysis.

What are the limitations of the Second Derivative Test?

The Second Derivative Test is limited in its applicability to functions that are twice differentiable. Additionally, it may not always provide a definitive result, especially for functions with complex shapes or multiple critical points. In these cases, further analysis or alternative methods may be needed to determine the nature of the critical points.

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