Verifying a point of infelction

[Andiio]

Is it sufficient to show that:
1. f''(x) = 0 and 2. f'(x) = 0

Or do you have to show:
1. f''(x) = 0, and 2. f'(x) is the same on either side of the P.O.I., i.e. concavity

Thanks!

[dc302]

You have to do the 2nd one. First one isn't true, for example, f(x) = x^4

[HarveyD]

Number 1 is only true for SPOIs right?

[dc302]

It is not true for any POIs or SPOIs, as I pointed out in my example.

If you take f(x) = x^4,

then f'(x) = 4x^3, f''(x) = 12x^2

and you can see that f'(0) = f''(0) = 0, even though it is a local minimum, not a POI.

[Zebra]

so how would you verify a SPOI?

[dc302]

You have to show that f'(a) = 0 and that f''(x) changes sign at x=a.

[abeybaby]

Is it sufficient to show that:
1. f''(x) = 0 and 2. f'(x) = 0

Or do you have to show:
1. f''(x) = 0, and 2. f'(x) is the same on either side of the P.O.I., i.e. concavity

Thanks!

showing that f'(x) is the same on either side isnt what you need.
you have to show that f''(x) changes sign, as dc pointed out

[dc302]

Is it sufficient to show that:
1. f''(x) = 0 and 2. f'(x) = 0

Or do you have to show:
1. f''(x) = 0, and 2. f'(x) is the same on either side of the P.O.I., i.e. concavity

Thanks!

showing that f'(x) is the same on either side isnt what you need.
you have to show that f''(x) changes sign, as dc pointed out

Oops I just realised what I said in my first post in this thread; read the opening post wrong. Yeah, showing f'(x) is the same sign on either side doesn't mean it's an inflection.

[Zebra]

my brains really fucked atm
so
spoi
f''(x)=0, f'(x)=0 (either side sign change)

poi
f'(x)=any value , f''(x)=0?

wtfwtwftwf

[dc302]

edit: sorry fcked it up:

SPOI:

f'(a) = 0
f''(x) changes sign at x=a


POI

f''(x) changes sign at x=a

[Zebra]

edit: sorry fcked it up:

SPOI:

f'(a) = 0
f''(x) changes sign at x=a


POI

f''(x) changes sign at x=a

SPOI?? f''(x)=0 NO?

[b^3]

Not always, as dc302 said above, it is true for y=x^4 which does not have a SPOI.

It is not true for any POIs or SPOIs, as I pointed out in my example.

If you take f(x) = x^4,

then f'(x) = 4x^3, f''(x) = 12x^2

and you can see that f'(0) = f''(0) = 0, even though it is a local minimum, not a POI.

[Zebra]

wow. this is new.
but if there is sign change on either side of f''(x)... mmm

oh! maybe i understand... maybe not...

[b^3]

For there to be a point of inflexion the function has to change from concave up to concave down are vice-versa. This means that the rate that the gradient is changing (i.e. f''(x)) must go from +ve to -ve or -ve to +ve. So that means the point inflects, it starts to bend back the other way (even though the gradient may stay +ve or -ve), if that makes sense.