Stationary Points of Inflection

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In summary: So, if there is a point of inflection, then the 2nd derivative will change signs. Now, given y=x^3 -9x^2+23x-16 on the interval [-3,7]the maximum and minimum values would be the turning points right?Yes, if you look at the graph. Also, how do i obtain an EXACT value for the x coordinates.There is no particular way to find an exact x-coordinate for a point of inflection. However, you can use the second derivative test to determine whether a given point is a local maximum or minimum.
  • #1
Cummings
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Now, given y=x^3 -9x^2+23x-16 on the interval [-3,7]

the maximum and minimum values would be the turning points right?

also, a stationary point of inflextion is where the grandient is zero, with a positive or negative gradient on both sides right?

i am asked to find the EXACT values of the x-coordinates of the points of inflection on the graph of -x^4 + 3x^3 + 5x^2 + 2x + 11

but, there are two maximums and one minimum, not a stationary point of inflection.

So, are minimum and maximum values points of inflections?

Also, how do i obtain an EXACT value for the x coordinates.
 
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  • #2
Cummings said:
Now, given y=x^3 -9x^2+23x-16 on the interval [-3,7]

the maximum and minimum values would be the turning points right?

also, a stationary point of inflextion is where the grandient is zero, with a positive or negative gradient on both sides right?

i am asked to find the EXACT values of the x-coordinates of the points of inflection on the graph of -x^4 + 3x^3 + 5x^2 + 2x + 11

but, there are two maximums and one minimum, not a stationary point of inflection.

So, are minimum and maximum values points of inflections?

Also, how do i obtain an EXACT value for the x coordinates.

You are mixing together the concepts "point of inflection" and "saddle point"

A "saddle point" is a point where the derivative is zero, but where the function does not achieve a local extremal value.

A "point of inflection" is a point where the curvature changes sign; in particular, the 2.derivative is zero at a "point of inflection".
 
  • #3
Cummings said:
Now, given y=x^3 -9x^2+23x-16 on the interval [-3,7]
the maximum and minimum values would be the turning points right?

Since you're given an interval, then you're dealing with local extrema here. I'm not sure what you mean by 'turning point'. I suggest you take a look at the definition of a local max. and min. again.

also, a stationary point of inflextion is where the grandient is zero, with a positive or negative gradient on both sides right?

i am asked to find the EXACT values of the x-coordinates of the points of inflection on the graph of -x^4 + 3x^3 + 5x^2 + 2x + 11

but, there are two maximums and one minimum, not a stationary point of inflection.

So, are minimum and maximum values points of inflections?

Also, how do i obtain an EXACT value for the x coordinates.

As arildno said, you're probably getting some of these concepts mixed up. There is a so-called second derivative test which you can use to find whether a given point on the function is a a local maximum or minimum. It basically says:

If for some p, f'(p) = 0 then
- If f''(p) > 0, there is a local minimum at (p, f(p))
- If f''(p) < 0, there is a local maximum at (p, f(p))
- If f''(p) = 0, then you're out of luck.

Hope that helps,
e(ho0n3
 
  • #4
arildno said:
You are mixing together the concepts "point of inflection" and "saddle point"

A "saddle point" is a point where the derivative is zero, but where the function does not achieve a local extremal value.

A "point of inflection" is a point where the curvature changes sign; in particular, the 2.derivative is zero at a "point of inflection".
Be careful with that. A point of inflection only occurs if the 2nd derivative changes signs, not simply if it is zero. For example, if [tex]f(x)=3x[/tex], then [tex]f''(x)=0[/tex] for all values of x, but there are clearly no inflection points.

But if you look at [tex]f(x)=x^3[/tex], [tex]f''(x)=6x[/tex] and we see that [tex]f''(0)=0, f''(-.1)=-.6, f''(.1)=.6[/tex] and the 2nd derivative is clearly changing signs at x=0, so there is an inflection point there.
 
  • #5
Hmm..that's what I meant with ("a point where the curvature changes sign"), but I see now that the last part of the sentence made the meaning ambiguous.
Thx.
 

What is a stationary point of inflection?

A stationary point of inflection is a point on a curve where the concavity changes. This means that the curve changes from being concave up to concave down (or vice versa) at this point.

How do you find stationary points of inflection?

To find stationary points of inflection, you must first find the second derivative of the function. Then, set the second derivative equal to zero and solve for the input value. This input value is the x-coordinate of the stationary point of inflection.

What is the significance of stationary points of inflection?

Stationary points of inflection can provide valuable information about the behavior of a function. They can indicate where the curvature of a curve changes, which can help determine the maximum and minimum points of a function.

Can a function have more than one stationary point of inflection?

Yes, a function can have multiple stationary points of inflection. This occurs when the second derivative of the function crosses the x-axis multiple times, indicating changes in concavity at multiple points.

How do stationary points of inflection differ from other types of stationary points?

Stationary points of inflection differ from other types of stationary points (such as maximum or minimum points) because they do not have a slope of zero. Instead, they have a slope of undefined or infinite due to the change in concavity.

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