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### Course: AP®︎/College Calculus AB>Unit 5

Lesson 7: Determining concavity of intervals and finding points of inflection: algebraic

# Mistakes when finding inflection points: second derivative undefined

Candidates for inflection points are points where the second derivative is zero *and* points where the second derivative is undefined. It's important not to overlook any candidate.

## Want to join the conversation?

• CAN someone please explain me what does sal mean when he says "not equal to zero, but where the second derivative equal to zero??"
• At , Sal means that there is an inflection point, not at where the second derivative is zero, but at where the second derivative is undefined. Candidates for inflection points include points whose second derivatives are 0 or undefined. A common mistake is to ignore points whose second derivative are undefined, and miss a possible inflection point.
Hope that I helped.
• Why is x a candidate for being an inflection point when g''(x) undefined?
• Think about how 0 is a critical point for something like |x|, even though the derivative at x=0 is undefined
• what if the question were to find the inflection point of the sq. root instead of the cube root and we had a situation where the second derivative function is not defined for all the numbers to the left of the inflection point candidate? How would one answer such a question?
• This example is a little tricky because x^(1/3) can be complex when x is negative (i.e., the equation y^3=x has three solutions for y--one real and two complex). I bring this up because some computer programs like Mathematica and Matlab plots x^(1/3) using their complex values when x<0...
• what does Sal mean by saying you have to be close enough so that nothing unusual happens? What could happen? I'm a little confused now.
• do we test for the x that was undefined to be defined in the original function when answering questions about inflection points?
• At why isn't x=0 a solution? You could determine that x=0 rendering the expression equal to zero.
(1 vote)
• X can't be equal to 0 because there is no x in the numerator which makes the numerator always equal to 2, if you try to plug in any number to get the second derivative equal to 0, the closest you can get is 0 in the denominator rendering the function undefined. Hope this helps
• How can one tell if a point of discontinuity (function is defined at that point) is a local minimum or maximum point?
• Points of discontinuity can be factored out and then you can check if its a local extrema the normal way. However, it can't actually be a local minimum or maximum because its not in the domain of the initial function.
(1 vote)
• Hello I am very confused here. So f(x) is obviously discontinuous at x=0, so how can there be an inflection point at x=0? By definition, aren't inflection points continuous? Thanks.
(1 vote)
• No. An inflection point is one where the second derivative changes between positive and negative. That doesn't require the function to be continuous.
• Can someone give me an example of a function where a point on it has a defined first derivative but an undefined second derivative? I'm having trouble understanding how a function with a point where only the second derivative is undefined would look.
(1 vote)
• Using an online graphing tool, try plotting something like
f(x) = {
-x^2/2 if x < 0
x^2/2 if 0 <= x
}

1st derivative:
f'(x) = {
-x if x < 0
x if 0 <= x
}

f'(0) from the right and from the left are both 0, so f'(0) = 0 and is defined.

2nd derivative:
f''(x) = {
-1 if x < 0
1 if 0 <= x
}

f'' from the left is -1, but f'' from the right is 1. Since these two aren't the same, f''(0) doesn't exist.