Whats the opposite/inverse of the point of inflection of a curve. Qualitatively it can be called a "point of diminishing returns", or described as the point where the slope of the curve ceases to increase at a previously sustained rate. What is the mathematical representation of this point?

thanks.

12/1/2005 10:28:48 AM

That sounds like you are describing an inflection point, that is the boundary between a curve that is concave up and one that is concave down.

12/1/2005 1:14:07 PM

but whether it is up or down its still called an "inflection point"?

12/1/2005 1:35:58 PM

yea

it is just where the second derivative is equal to zero

12/1/2005 1:45:32 PM

ok thanks

12/1/2005 4:44:08 PM

This might be obvious, I don't know, but it's not quite enough to say that inflection points are where the second derivative is zero--The second derivative also needs to change signs across that point.

For example, f(x)=x^4 has second derivative zero at x=0, but this is not an infection point since the function is concave up everywhere (the second derivative is never negative).

12/1/2005 5:29:10 PM

so the opposite would be where the 2nd deriv = infiniti?

12/1/2005 9:42:18 PM

A line Y=mx+b has y''=0 everywhere but is neither concave up or down anywhere. Or is it both... hmm...

12/1/2005 10:17:33 PM

for the 2nd derivative rule to work, it has to be a newly defined zero--as in, if the first derivative is also zero at that point, that point can't also be a point of inflection

12/2/2005 12:38:22 PM

^wrong.
Cabbage had it right--end of thread.

12/2/2005 4:54:30 PM

if that was the end then how are you reading this?

12/2/2005 6:00:41 PM

^^no, it isn't wrong.

Cabbage isn't wrong either.

12/4/2005 12:04:45 PM

^ and ^^^^ what about f(x)=x^3 ? Notice that

f ' (x) = 3x^2
f '' (x) = 6x

Clearly f ' (0) = 0 and (0,0) is a point of inflection since f '' (x) < 0 when x < 0
and f '' (x) > 0 when x > 0. So here is an example of an inflection point which is not a newly defined zero. Your statement needs some adjustment.

12/4/2005 12:32:58 PM

^^ .
I hope you're not a math major, or anything science related.
And as mathman pointed out, the counter example to your claim is trivial. I thought you would think about it after I said you were wrong.

Quote :

"Your statement needs some adjustment."

that's an understatement.

12/4/2005 2:35:35 PM