Point of inflection, max and min?

[apple2357]

I know that all cubics have a point of inflection and many can have Max and min stationary points.
Is it possible for any polynomial to have a Max/min and a Stationary point of inflection though? Can anyone think of function with this property? It would have to be a quartic or above I am assuming but not sure if it even exists? I have been playing around but not getting anywhere?

 

[Harry_the_cat]

I know that all cubics have a point of inflection and many can have Max and min stationary points.
Is it possible for any polynomial to have a Max/min and a Stationary point of inflection though? Can anyone think of function with this property? It would have to be a quartic or above I am assuming but not sure if it even exists? I have been playing around but not getting anywhere?

Not all cubics have a horizontal point of inflection. At most they have two stationary points. Quartics have at most three stationary points.
Yes a polynomial can have a max/min point and a PHI. Try to draw one.

 

[HallsofIvy]

At a "max" or "min" or "inflection point" of a polynomial the first derivative must be 0. At an inflection point the second derivative is also 0. So lets see if we can create a polynomial function that has a "max" at x= -1, and "inflection point" at x= 0, and a "min" at x= 1.

We must have f'(-1)= f'(0)= f'(1)= 0, f''(-1)< 0, f''(1)> 0 and f''(0)= 0. To satisfy those six conditions we need six coefficients to determine so we need at least a fifth degree polynomial, $\displaystyle f(x)= ax^5+ bx^4+ cx^3+ dx^2+ex+ f$.

Then $\displaystyle f'(x)= 5ax^4+4bx^3+ 3cx^2+ 2dx+ e$ and $\displaystyle f''= 20ax^3+ 12bx^2+ 6cx+ 3d$.

We must have $\displaystyle f'(-1)= 5a- 4b+ 3c- 2d+ e= 0$, $\displaystyle f'(0)= e= 0$, $\displaystyle f'(1)= 5a+ 4b+ 3c+ 2d+ e= 0$, and $\displaystyle f''(0)= 3d= 0$ as well as $\displaystyle f''(-1)= -20a+ 12b- 6c+ 3d> 0$ and $\displaystyle f''(1)= 20a+ 12b+ 6c+ 3d< 0$. Assign whatever values you want for f(-1), f(0), and f(1) so that you have six equations to solve for a, b, c, d, e, and f.

 

[apple2357]

At a "max" or "min" or "inflection point" of a polynomial the first derivative must be 0. At an inflection point the second derivative is also 0. So lets see if we can create a polynomial function that has a "max" at x= -1, and "inflection point" at x= 0, and a "min" at x= 1.

We must have f'(-1)= f'(0)= f'(1)= 0, f''(-1)< 0, f''(1)> 0 and f''(0)= 0. To satisfy those six conditions we need six coefficients to determine so we need at least a fifth degree polynomial, $\displaystyle f(x)= ax^5+ bx^4+ cx^3+ dx^2+ex+ f$.

Then $\displaystyle f'(x)= 5ax^4+4bx^3+ 3cx^2+ 2dx+ e$ and $\displaystyle f''= 20ax^3+ 12bx^2+ 6cx+ 3d$.

We must have $\displaystyle f'(-1)= 5a- 4b+ 3c- 2d+ e= 0$, $\displaystyle f'(0)= e= 0$, $\displaystyle f'(1)= 5a+ 4b+ 3c+ 2d+ e= 0$, and $\displaystyle f''(0)= 3d= 0$ as well as $\displaystyle f''(-1)= -20a+ 12b- 6c+ 3d> 0$ and $\displaystyle f''(1)= 20a+ 12b+ 6c+ 3d< 0$. Assign whatever values you want for f(-1), f(0), and f(1) so that you have six equations to solve for a, b, c, d, e, and f.

thanks, I will get to work on this!

 

[apple2357]

I experimented using graphical software and it appears there are plenty.

Here is one, if anyone is interested:

y= -x^5+3x^3

I guess a quintic is the lowest order polynomial to have this property.

 

[Steven G]

I experimented using graphical software and it appears there are plenty.

Here is one, if anyone is interested:

y= -x^5+3x^3

I guess a quintic is the lowest order polynomial to have this property.

Yes, there are 'plenty'. If you find some f(x) that works, then g(x) = f(x) + k, k is any real number, will also work. Raising or lowering a function will still have the poi and max/min at the same x-values.

 

[Harmless Drudge]

I know that all cubics have a point of inflection and many can have Max and min stationary points.
Is it possible for any polynomial to have a Max/min and a Stationary point of inflection though? Can anyone think of function with this property? It would have to be a quartic or above I am assuming but not sure if it even exists? I have been playing around but not getting anywhere?

If you mean, 'What is the lowest order polynomial function that can have both an absolute max/min and an inflection point?' then, yes, it would be a quartic. Just thinking intuitively for a moment, it would have to be an even number exponent, otherwise the end behaviours would diverge, one toward infinity, and the other toward negative infinity. It would also have to be at least a cubic, in order for its concavity to change, thus exhibiting an inflexion point.

The lowest order polynomial in which this is possible is a quartic, though not all quartics exhibit inflexion. Take a naked quartic like y = x^4 as an example. It simply starts at the origin and has nowhere to go but up, as none of the lower order exponents in x challenge the upward tendency of the function as it progresses leftward or rightward in x.

As far as enumerating how many functions exhibit both absolute max/min and inflexion, it is at least a simple aleph nought infinity. We can demonstrate this just with quartics. Ax^4 + Bx^3 + Cx^2 + Dx + E can have any of an infinite number of values for E.

Perhaps a disciple of Georg Cantor can enlighten us as to whether the number of possibilities is of a higher infinity.

Or, of course, I might be misunderstanding your question entirely.

 

[apple2357]

Yes, there are 'plenty'. If you find some f(x) that works, then g(x) = f(x) + k, k is any real number, will also work. Raising or lowering a function will still have the poi and max/min at the same x-values.

I can't see why a quartic can't have this property? When you differentiate it , it's possible to have three solutions? Or does the second derivative cause a problem?