Point of inflection - always halfway between 2 critical points?

In summary, the point of inflection is the point where the concavity of a function changes from up to down or down to up. It is always equidistant from two critical points, which are points where the function changes from increasing to decreasing or decreasing to increasing. For cubics, there can be at most two critical points and one inflection point, which can only lie on the y-axis at x=0 due to its unicursal nature.
  • #1
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point of inflection -- always halfway between 2 critical points??

when you look at the graph of a function
say, f(x) = 5x^3 - 2x^2 + 3x - 1

will the point(s) of inflection always be equidistant from 2 critical points (ie the 2 nearest critical points)?


point of inflection -- point where the concavity changes from up to down / down to up

critical point -- point where the function changes from increasing to decreasing or decreasing to increasing
 
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  • #2
Okay, this is about cubics. With any cubic you can do a linear transformation and produce a coordinate system in which the curve
a) passes through the origin and
b) is unicursal, it only passes through each coordinate line once, and
c) the part in the negative half of the plane is the reversed mirror image of the part in the positive half of the palne.

Now the cubic has at most two critical points and one inflection because its first derivative is a quadratic with two or no real roots, and its second derivative is linear with one root.

So take that one and only one inflection point. Where can it lie? Not in the negative half of the plane, because there's no second inflection point to match it on the positive side. And by the same reasoning not in the right half plane either. Therefore it lies on the y-axis, at x=0.

I think I'll leave the rest of the proof for you to finish.
 
  • #3


No, the point of inflection is not always halfway between two critical points. While it is true that the point of inflection is where the concavity changes, it is not limited to being exactly halfway between two critical points. In fact, the point of inflection can occur at any point where the concavity changes, regardless of its distance from the nearest critical points. This is because the point of inflection is determined by the second derivative of the function, which can change at any point where the first derivative is zero. Therefore, the location of the point of inflection is not directly related to the location of critical points.
 

1. What is a point of inflection?

A point of inflection is a point on a curve where the direction of the curve changes from concave upward to concave downward (or vice versa). It is also known as a point of change in curvature.

2. How is a point of inflection different from a critical point?

A critical point is a point on a curve where the derivative is equal to zero or undefined, while a point of inflection is a point where the second derivative is equal to zero or undefined. In other words, a critical point indicates a potential maximum or minimum on the curve, while a point of inflection indicates a change in the curvature of the curve.

3. Why is a point of inflection always halfway between two critical points?

This is because at a point of inflection, the second derivative of the curve is equal to zero, meaning that the curve is neither concave up nor concave down. Therefore, the curve must be changing from concave up to concave down (or vice versa), and this change occurs at the halfway point between two critical points.

4. Can a point of inflection exist without any critical points?

Yes, it is possible for a curve to have a point of inflection without any critical points. This occurs when the curve is a straight line, as the second derivative of a straight line is always equal to zero.

5. How can we identify a point of inflection on a graph?

To identify a point of inflection on a graph, we can look for a change in the concavity of the curve. This can be done by finding the second derivative of the curve and setting it equal to zero. If there is a change in sign of the second derivative at that point, it is a point of inflection.

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