Condition for inflection point

In summary, an inflection point is where the curve changes its concavity. The condition for this is that the second derivative should be zero, but this alone is not sufficient as seen in the example of y=x^4. Additional conditions involve the odd derivatives also being zero, which may seem confusing. However, this can be understood by considering that the sign of the second derivative determines the concavity of the curve, and if this sign changes at a point, the second derivative must also be zero at that point. The same concept applies for higher odd derivatives. This can be further illustrated with the Taylor series, where the first non-zero derivative determines the shape of the curve. If this derivative is an odd number, there will be an inf
  • #1
jd12345
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Well i know that an inflection point is where the curve changes its concavity.
But i don't really understand the conditions for it.
It says that second derivative should be zero(but that's not sufficient). I understand this. Second derivative being zero is not sufficient, example is y =x^4. So further condition is that some of the following odd derivatives should be zero which i don't understand

What's with the odd derivative? I don't get the intuition
 
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  • #2
concavity is measured buy the sign of the second derivative, so changing concavity means the second derivative changes sign.

If the second derivative changes sign at c, then in particular it has to be zero at c (if it exists there), but not vice versa. But if the third derivative is non zero at c, then the second derivative was either increasing or decreasing at c. Thus if the second derivative was zero at c and also the third derivative was not zero, then the second derivative must have changed sign...the same game goes on...A simple illustration is to think of the Taylor series. Your curve looks locally like the lowest non zero term of the Taylor series. So If the 5th derivative is the first non zero derivative then your curve looks like y = x^5, which has an inflection point,

but if the first non zero derivative is the 6th, then your curve looks like y = x^6, which has no inflection point.
 

1. What is an inflection point?

An inflection point is a point on a graph where the curve changes from concave to convex or vice versa. This means that the slope of the curve changes from increasing to decreasing or vice versa. It is also known as a point of inflection or a flex point.

2. How do you determine the condition for an inflection point?

The condition for an inflection point is when the second derivative of the function is equal to zero or does not exist. In other words, the slope of the slope must change sign or become undefined at that point on the curve.

3. Why is the condition for an inflection point important?

Inflection points are important because they indicate a change in the concavity of a curve. This can help us understand the behavior and shape of a function, and can also be used to find critical points and extrema.

4. Can a function have more than one inflection point?

Yes, a function can have multiple inflection points. This occurs when the second derivative of the function changes sign or becomes undefined at multiple points on the curve. These points will be where the concavity of the curve changes.

5. How do inflection points relate to real-world phenomena?

Inflection points can be seen in many real-world phenomena, such as changes in population growth, stock market trends, and weather patterns. They can also be used to analyze and predict the behavior of complex systems, making them an important concept in many fields of science and engineering.

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