properties of IP and HIP (2 Viewers)

[tazzyboys]

so basically i know for an IP, f'' = 0 and there must be a concavity change.
i dont understand the concavity change and how i would test for it
im familiar with a table technique, but not exactly sure how it goes.

for a HIP, f' and f'' must = 0.
does there need to be a change in concavity?


what does concavity change mean?
different signs on either side of the point?
so is + 0 + a change in concavity?

 

[cwag]

i assume u mean points of inflection and horizontal points of inflection.

to find stat points let y' = 0 right? then to find nature of these stat points sub the x values you found for y'=0 into y''. if y'' >0 its a local minimum, if y''< 0 its a local maximum. However, if y''=0 then it must be tested. i usually test in either first derivative or second, depending on which is easier to calculate. If you have a change in sign when you test either side for the first derviate then the gradient changes, so u do not have a HPI it will be either \_/ or /\. If there is a change in sign for second derivative then u do have a HPI (because concavity has changed)

e.g

y=x⁴
y' = 4x³
therefore y'=0 when x=0

to find nature.
y"=12x²
sub x=0, y"=0. we must test

first derivative test.. x=-1 y'= -4, x=1, y'=4....there is a change in sign so it must go \_/

OR second derivative test....x=-1, y''=12, x=1, y''=12.....no change in sign, therefore concavity must go from positive, to 0, then to positive again. \_/.

and we know y=x⁴ looks like a parabola with vertex (0,0)


to just find a point of inflection. let y''=0 no need for testing, if y' does not =0


probably confused you more....im sorry

 

[lyounamu]

Can I briefly ask something here?

When you find the point of inflexion. Do you have to write the test as well?

 

[haywattsngu]

I'm assuming that you've done y"=0 and found values for it. You should check those values on the paper, i.e. whether they are points of inflexion indeed, just to be on the safe side. Marks can, and usually are, lost if those obvious things are missing.
It's not that hard anyway. And it also removes those post-exam doubts of whether you did something right or wrong!

 

[cwag]

Can I briefly ask something here?

When you find the point of inflexion. Do you have to write the test as well?

no, you don't....unless y' also equals 0 at that point.