EXAM 2 QUESTIONS

[dc302]

oh boy... what do you guys mean by 'f''(x) changes sign at x'?

f''(x) should be zero!


so...... oh god i've been looking for how to verify POI / SPOI for like the last few days but I still don't get it.
ARGH!

Well, for f''(x) to change signs at x=a, we need that f''(x) is positive on one side, negative on the other side, and of course 0 at x=a.

Ie.

y = x^3, y' = 3x^2, y'' = 6x

We want y'' to change signs at x = 0

So, y''(0) = 0, y''(1) = 6 > 0, y''(-1) = -6 < 0. So thus it changes sign and we have a POI. I may have been unclear but to 'change signs at x=a' automatically means it must be 0 at x=a anyway. 

[Zebra]

so we are only considering the sign changes in f''(x) not f'(x)?

sorry i'm really struggling

[abeybaby]

so we are only considering the sign changes in f''(x) not f'(x)?

sorry i'm really struggling

thats correct. what f'(x) does around x=a doesnt mean anything for points of inflexion

[Mao]

so we are only considering the sign changes in f''(x) not f'(x)?

sorry i'm really struggling

thats correct. what f'(x) does around x=a doesnt mean anything for points of inflexion

Extension:
To see if f''(x) changes sign, you can simply evaluate f'''(x). If , then it changes sign. If it does not, then the third derivative test for point of inflexion is inconclusive.

[TrueTears]

also just to emphasize the point, plugging in values and checking whether f''(x) changes sign is a good idea in practice however to be more pedantic, evaluating f'''(x) would be a bulletproof method, this is because if you plug in values either side of 0, you might select "too big" of a number and thus lead to incorrect conclusions.

[Mao]

also just to emphasize the point, plugging in values and checking whether f''(x) changes sign is a good idea in practice however to be more pedantic, evaluating f'''(x) would be a bulletproof method, this is because if you plug in values either side of 0, you might select "too big" of a number and thus lead to incorrect conclusions.

but if f'''(x) turns out to be inconclusive, we'll end up having to evaluate or higher!

Here, have the general n-order derivative test for shits and giggles.

http://en.wikipedia.org/wiki/Higher-order_derivative_test

[TrueTears]

LOL, i'd rage if they did that in a VCE paper haha

[Mao]

LOL, i'd rage if they did that in a VCE paper haha

... , therefore is a stationary point of inflexion.



We should probably bring it back to Specialist level now.

[dc302]

I'm pretty sure you're not allowed to use the third derivative though, as they haven't proved it in the course.

[Zebra]

:S daaa fuck... argh exam soon! gonna get some sleeep!