Calculus inflection point question

In summary, the given conversation discusses the existence of an inflection point at x = 0 for a function f with a continuous second derivative. It is stated that in order for f'' to have an inflection point, it needs to change sign near that point. However, the person is unsure if there is a test involving f' to determine this. They suggest that it may be impossible to determine if f'' changes sign without analyzing the neighborhood of x = 0. A function example, f(x) = 1+2x^2, is also mentioned for further understanding.
  • #1
daysrunaway
19
0

Homework Statement


Suppose that f has a continuous second derivative for all x, and that f(0) = 1, f'(0) = 2, and f''(0) = 0. Does f have an inflection point at x = 0?


Homework Equations


none


The Attempt at a Solution


I know that for f'' to have a point of inflection, it needs to change sign near that point, but I can't remember if there's a test for f'' that involves f' to find out if it does. Should I just say that it is impossible to say, since I can't analyze the 'neighborhood' of x = 0 to see if f'' changes sign?
 
Physics news on Phys.org
  • #2
Since f' is continuous for all x, small changes in x correspond to small changes in f'(x). For a value c very close to 0, f'(c) would have to decrease by 2 units in order for it to become negative.
 
  • #3

FAQ: Calculus inflection point question

What is a calculus inflection point?

An inflection point is a point on a curve where the concavity (or direction of curvature) changes. In other words, it is the point where a curve changes from convex to concave or vice versa.

How do you find the inflection point of a function?

To find the inflection point of a function, you need to find the second derivative of the function and set it equal to zero. Then, solve for the x-value at which the second derivative is equal to zero. This x-value is the inflection point.

What is the significance of an inflection point in calculus?

An inflection point is significant because it represents a change in the behavior of a function. It is where the rate of change of the function changes, and can indicate points of maximum or minimum values.

Can a function have multiple inflection points?

Yes, a function can have multiple inflection points. In fact, any polynomial function of degree 3 or higher can have multiple inflection points.

How can calculus inflection points be applied in the real world?

Inflection points can be applied in real world situations where there is a change in direction or behavior. For example, in economics, inflection points can represent points of market saturation or shifts in consumer behavior. In physics, inflection points can indicate changes in the direction of a moving object.

Similar threads

Inflection Point Calculation: Reduction of Cubic with Second Derivative Method
Replies
1
Views
787
Graphing Derivatives: Concavity and Inflection Points
Replies
4
Views
1K
Determine the values of a and b that have inflection points
Replies
18
Views
2K
Queries regarding Inflection Points in Curve Sketching
Replies
2
Views
2K
Problem 1-23 and 1-24 from Spivak's Calculus on Manifolds
Replies
6
Views
1K
Using integral calculus to find the equation of the quartic
Replies
5
Views
2K
Prove that solutions to autonomous first order DE can't have local maxima
Replies
2
Views
859
Finding the Inflection points of x^2-4√x
Replies
4
Views
1K
Exploring the Sign of a 2nd Derivative Function
Replies
7
Views
1K
Inflection Points and Intervals
Replies
1
Views
2K

Hot Threads

Recent Insights

Back
Top