Point of Inflection question (1 Viewer)

[no_arg]

Here's an odd one!
Define f to be the piecemeal function f(x)=x^2 for x>=0 and
f(x) =-(x^2) for x<=0

The graph of f is then sort of like x^3

Does f have a point of inflection at the origin?

 

[conics2008]

the graph looks SIMILAR to x^3 but NOOOOOOOOOOOOOOO

because x^2 dy/dx = 2x d^2y/dx^2=2 hence no point of inflexion ??

 

[buchanan]

It changes concavity at (0,0), so this is an inflection.

But all the derivatives of order greater than 1 do not exist at (0,0).

 

[no_arg]

Is only a change in concavity required?

 

[buchanan]

Some books require it to be smooth as well, and your example qualifies for this, so is not too controversial.

However other examples where it is not smooth at (a,b) may also change concavity at (a,b). This is where it gets controversial. Some books still call it an inflection, others don't.

For example,

f(x)=-x(x+1) for x &le; 0

f(x)=x² for x &gt; 0

This changes concavity at (0,0). But it is not smooth at (0,0).

Is (0,0) an inflection? Depends on the definition.

 

[buchanan]

But getting back to your example, I think it is a good one because the syllabus says y'' must be 0 at an inflection and change sign either side. Whilst that's OK for most examples, there will be some examples (like yours) for which it isn't true. So the syllabus's description does not suffice as a definition, yet is quite commonly used, eg., in

Bronshtein, I. N. and Semendyayev, K. A. Handbook of Mathematics, 4th ed. New York: Springer-Verlag, 2004, p. 231.

Their definition won't suffice for the example in

Purcell, E. J. and Varberg, D., Calculus with Analytic Geometry, 5th ed. Prentice-Hall, p. 166

(attached below)

nor for yours.

Purcell and Varberg just define an inflection as a point at which concavity changes. They don't have to be smooth and y' and y'' don't have to exist. The curve must however be continuous at an inflection point.

 

[ronnknee]

My Math teacher always tells us f''(x) = 0 is necessary but not sufficient proof for a point of inflexion. It must also change in concavity.

In this case, yes it does change in concavity but f''(x) does not equal to 0. Therefore there is no point of inflexion

 

[buchanan]

Your teacher may not be aware that there are more than 1 definition.

You'll get a different answer to no_arg's question depending on which definition you use.