Inflection point confusion

In summary, the intervals of concavity for the function f(x) = x^{4} - 2x^{2} + 3 are (-\infty, -\frac{1}{\sqrt{3}}) \cup (\frac{1}{\sqrt{3}}, \infty). The inflection points are located at \left(1/\sqrt{3}, 22/9\right) and \left(-1/\sqrt{3}, 22/9\right).
  • #1
PsychStudent
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Homework Statement


[tex] f(x) = x^{4} - 2x^{2} + 3[/tex]

Find the intervals of concavity and the inflection points.

Homework Equations


[tex]f''(x) = 4(3x^{2}-1)[/tex]

The Attempt at a Solution


[tex]f''(x)[/tex] is zero at [tex]\pm\frac{1}{\sqrt{3}}[/tex]
I've found the correct intervals of concavity, which are [tex](-\infty, -\frac{1}{\sqrt{3}}) \cup (\frac{1}{\sqrt{3}}, \infty)[/tex]
I would expect the inflection points to be [tex]\pm\frac{1}{\sqrt{3}}[/tex], which is partly correct, but the answer the book gives is [tex]\pm\frac{1}{\sqrt{3}}, \frac{22}{9}[/tex].
I can't see how [tex]\frac{22}{9}[/tex] could be an inflection point. It does not equal zero when plugged into the second derivative.

Thanks
 
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  • #2
22/9 isn't an "inflection point" but then neither are [itex]1/\sqrt{3}[/itex] nor [itex]1/\sqrt{3}[/itex]! They are not points! What your book is saying is that [itex]\left(1/\sqrt{3}, 22/9\right)[/itex] and [itex]\left(-1/\sqrt{3}, 22/9\right)[/itex] are the inflection points. That is, when x is [itex]1/\sqrt{3}[/itex] or [itex]-1/\sqrt{3}[/itex], y is equal to 22/9.
 
  • #3
HallsofIvy said:
22/9 isn't an "inflection point" but then neither are [itex]1/\sqrt{3}[/itex] nor [itex]1/\sqrt{3}[/itex]! They are not points! What your book is saying is that [itex]\left(1/\sqrt{3}, 22/9\right)[/itex] and [itex]\left(-1/\sqrt{3}, 22/9\right)[/itex] are the inflection points. That is, when x is [itex]1/\sqrt{3}[/itex] or [itex]-1/\sqrt{3}[/itex], y is equal to 22/9.

Of course! Thanks for your help
 

What is an inflection point?

An inflection point is a point on a curve where the direction of the curve changes, either from concave up to concave down or vice versa.

How can an inflection point be identified?

An inflection point can be identified by looking at the second derivative of the curve. If the second derivative changes sign at a specific point, that point is an inflection point.

What is the significance of an inflection point?

Inflection points are important in mathematical and scientific analysis because they indicate a change in the behavior of a system. They can also be used to find the maximum or minimum points of a curve.

What causes confusion about inflection points?

One common cause of confusion is mistaking an inflection point for a maximum or minimum point. Additionally, inflection points can be difficult to identify on complex curves with multiple inflection points.

How can inflection points be used in real-life applications?

Inflection points are used in various fields, such as economics, physics, and biology, to analyze and predict changes in systems. They can also be used in data analysis to identify patterns and trends in data.

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