How to verify an oblique point of inflection?

[ahat]

Because a classical sign test isn't going to work - there isn't a stationary point! D:

To clarify, I mean letting the second derivative = 0, and then finding the value. How do we show that it is an oblique point of inflection. Do we have to show that the 'concavity changes'?

Thanks!

[abeybaby]

Im assuming you mean 'non-stationary point of inflection' when you say 'oblique point of inflection'. A point of inflection is a point where concavity changes, as you said. that means you'd need to see a sign change in the 2nd derivative to verify that something is a POI. to find points which potentially have a sign change, set the 2nd derivative to 0. Make sure you check to see if there is actually a sign change in the 2nd derivative to verify that it is a POI.

Say (a, f(a)) is a POI. if f'(a)=0, then (a,f(a)) is called a stationary POI. If f'(a) =/= 0, then (a,f(a)) is just a POI, which is what I assume you meant by oblique POI?

[ahat]

Im assuming you mean 'non-stationary point of inflection' when you say 'oblique point of inflection'.

Yes, thanks My question though, for a stationary point, we could do a sign test to  show that it is a maximum/minimum/stationary point of inflection.

How do we show, what test can we do, to show that a non-stationary point of inflection (i.e. when f"(x) = 0) is in fact, a "non-stationary point of inflection." (2006 Exam 2, 4 cii for reference).

[abeybaby]

do you mean: How can we show that there is a POI at x=a?

or do you mean: given that there is a POI at x=a, show that the POI is non-stationary?

[lzxnl]

Let's make this simple.
What is a point of inflection in general? It is where the second derivative changes sign. AKA if the second derivative is continuous, which in VCE cases it generally is, it must equal zero at the point of inflection.
There are cases, however, when then second derivative is zero but it doesn't change sign. The only way that is possible is if we have a cusp like a mod graph or we have a smooth turning point, like a parabola. Again, in VCE, you're not likely to come across the former, so let's look at the latter.
A smooth turning point of a graph occurs when the derivative is zero. Now, we're considering a second derivative graph, and we want to make sure that the x-intercept is not a turning point. Let the function f(x) have a supposed point of inflection at (a, f(a)). Then, to make sure that the x-intercept of y=f''(a) is not a turning point, we find f'''(a), a third derivative, but merely the derivative of the second derivative. If f'''(a) is zero, you could potentially have a turning point of the second derivative function. If it isn't, you know FOR SURE that (a, f(a)) is a point of inflection. If f'''(a) is zero...you're just unlucky then you're probably then better off checking to see if f''(a) changes sign. Calculator

[BasicAcid]

Let's make this simple.
What is a point of inflection in general? It is where the second derivative changes sign. AKA if the second derivative is continuous, which in VCE cases it generally is, it must equal zero at the point of inflection.
There are cases, however, when then second derivative is zero but it doesn't change sign. The only way that is possible is if we have a cusp like a mod graph or we have a smooth turning point, like a parabola. Again, in VCE, you're not likely to come across the former, so let's look at the latter.
A smooth turning point of a graph occurs when the derivative is zero. Now, we're considering a second derivative graph, and we want to make sure that the x-intercept is not a turning point. Let the function f(x) have a supposed point of inflection at (a, f(a)). Then, to make sure that the x-intercept of y=f''(a) is not a turning point, we find f'''(a), a third derivative, but merely the derivative of the second derivative. If f'''(a) is zero, you could potentially have a turning point of the second derivative function. If it isn't, you know FOR SURE that (a, f(a)) is a point of inflection. If f'''(a) is zero...you're just unlucky then you're probably then better off checking to see if f''(a) changes sign. Calculator

How do you tell if the point of inflexion is a stationary point of inflexion or not though?

[lzxnl]

The name should make it self-evident. A stationary point of inflection has f'(a)=0 as well as the graph of y=f(x) is stationary at that x value.

[sin0001]

Yes, thanks My question though, for a stationary point, we could do a sign test to  show that it is a maximum/minimum/stationary point of inflection.

How do we show, what test can we do, to show that a non-stationary point of inflection (i.e. when f"(x) = 0) is in fact, a "non-stationary point of inflection." (2006 Exam 2, 4 cii for reference).

What I did for that question was show that d^2y/dx^2 >0 for when y < ½, and is <0 for when y > ½

[ahat]

do you mean: How can we show that there is a POI at x=a?

or do you mean: given that there is a POI at x=a, show that the POI is non-stationary?

So, for the VCAA question I referenced, if I found the THIRD derivative and subbed in the point and it didn't equal 0 (it didn't by the way ) then that would be sufficient to verify?

The name should make it self-evident. A stationary point of inflection has f'(a)=0 as well as the graph of y=f(x) is stationary at that x value.

Is that sufficient?

[brightsky]

to find points of inflection in general, let f''(x) = 0. solve for x. just say you get x = a. THEN, check concavity on both sides of x=a. if there is a change in concavity, then x=a is a point of inflection. now, there are two types of points of inflection: non-stationary and stationary. to determine which one occurs at x=a, simply find f'(a). if f'(a) = 0, then x=a is a stationary point of inflection. if f'(a) does not equal to 0, then x=a is a non-stationary point of inflection.

avoid confusion at all costs. point of inflection = point at which concavity changes sign. work with that definition. what you need to show will become immediately obvious once you truly understand that definition.

[ahat]

to find points of inflection in general, let f''(x) = 0. solve for x. just say you get x = a. THEN, check concavity on both sides of x=a. if there is a change in concavity, then x=a is a point of inflection. now, there are two types of points of inflection: non-stationary and stationary. to determine which one occurs at x=a, simply find f'(a). if f'(a) = 0, then x=a is a stationary point of inflection. if f'(a) does not equal to 0, then x=a is a non-stationary point of inflection.

avoid confusion at all costs. point of inflection = point at which concavity changes sign. work with that definition. what you need to show will become immediately obvious once you truly understand that definition.

Thankyou. Concise - perfect.