point of inflection

[brenny]

if a question asks find where there is a point of inflection what is needed to be sufficient and when they ask for a stationary point of inflection what is needed to prove this?

[/0]

For a point of inflection you need to find where the second derivative is 0.
For a stationary point of inflection you need to find where the second derivative and first derivative are both 0.

Technically for both you need to show that the second derivative changes sign at those points, but I haven't seen any papers where this has been required. If you want to be safe you should also do this.

[brenny]

and a maximum/minimum turning point is considered a point of inflection but not stationary yes?

[/0]

No, they're not points of inflection!

In general,

Max turning points occur when and

Min turning points occur when and

Points of inflection occur when

There are exceptions to these rules but they are very rare

[brenny]

oh ok. yeh i do always let dy/dx=0 and then use second order deriviative to find its nature just i read on wikipedia

Isolated stationary points of a C1 real valued function  are classified into four kinds, by the first derivative test:

 
Saddle points (coincident stationary points and inflection points). Here one is rising and one is a falling inflection point.a minimal extremum (minimal turning point or relative minimum) is one where the derivative of the function changes from negative to positive;
a maximal extremum (maximal turning point or relative maximum) is one where the derivative of the function changes from positive to negative;
a rising point of inflection (or inflexion) is one where the derivative of the function is positive on both sides of the stationary point; such a point marks a change in concavity
a falling point of inflection (or inflexion) is one where the derivative of the function is negative on both sides of the stationary point; such a point marks a change in concavity

i thought max/min t.p are point of inflection not stationary point of inflection where a staionary point of inflection is when deriviate goes from + to 0 to + or - to 0 to -

just clarifying before the exam

[addikaye03]

For inflection points, you not only set d^2(y)/dx^2=0 but you also test for change in concavity, this is done by test two points, one before and one after the point at which y''=0. If changes then it's an inflection.

Another thing that many people don't know is INSTEAD of testing values before and after y''=0, you can use the THIRD DERIVATIVE TEST.

THIRD DERIVATIVE TEST:

If y''=0 (i.e you find a value of the variable, usually x)

and then y''' =/= 0 (doesn't equal 0)

then point x is an inflection point.

[humph]

For inflection points, you not only set d^2(y)/dx^2=0 but you also test for change in concavity, this is done by test two points, one before and one after the point at which y''=0. If changes then it's an inflection.

Another thing that many people don't know is INSTEAD of testing values before and after y''=0, you can use the THIRD DERIVATIVE TEST.

THIRD DERIVATIVE TEST:

If y''=0 (i.e you find a value of the variable, usually x)

and then y''' =/= 0 (doesn't equal 0)

then point x is an inflection point.

That's pretty pointless as there are lots of functions whose third derivative is also zero, so you have to use the fourth derivative test. But then the fourth derivative might also be zero... E.g. has for all , and it is only .

Moral of the story: just check either side of the point.