Question with maximum points of some curves...

Discuss the language of the universe.

Moderators: Calilasseia, ADParker

Question with maximum points of some curves...

#1  Postby LjSpike » Apr 11, 2017 3:54 pm

https://www.desmos.com/calculator/fyibg1b1sb

Is there a curve which links the maximum and minimum points of those lines like the black line joins them up, but obviously, a curve rather then dot-to-dot?

Ditto for the other maximum and minimum points (those not joined up by a line).
LjSpike
THREAD STARTER
 
Posts: 89
Age: 20
Male

United Kingdom (uk)
Print view this post

Ads by Google


Re: Question with maximum points of some curves...

#2  Postby Thommo » Apr 11, 2017 4:55 pm

There are always infinitely many curves that link any finite set of points. Any set of 2 points will lie on exactly one polynomial of degree 1 and infinitely many polynomials of higher degree. Any set of 3 points will lie on exactly one polynomial of degree 2 and infinitely many polynomials of higher degree. And so on.

You can easily find these solutions by solving simultaneous equations for a polynomial of suitable degree.

There will also be infinitely many other solutions for various curve types and sets of points, including many smooth curves or functions (e.g. fitting logarithmic or exponential curves to sets of two points is often easy).

-----

However, I've written that because it's interesting, I'm not sure it's what you want.

You've plotted a few examples of a family of curves, which have the function f(x)=(x+k)(x+2k)(x+2.5k).

Expanding that out: f(x) = x3 + 5.5kx2 + 9.5k2x + 5k3

Differentiate to find the turning points: f'(x) = 3x2 + 11kx + 9.5k2

Find the solutions by putting that into the quadratic formula with a = 3, b = 11k, c = 9.5k2 gives two solutions of x = (-11k + SQRT(7)k)/6 and (-11k - SQRT(7)k)/6 we can note that you've simply plotted the turning points stemming from the second solution by inspection.

This allows us to relate x and k by the formula x = (-11 - SQRT(7))k/6 or x = -2.274k (taking k to 3 d.p). Or equivalently k = -x/2.274.

Finally we can substitute that back into f(x) to get the curve of the solution family as g(x) = (0.560x)(0.121x)(-0.099x) = -0.006706415x3.

To answer the follow-on question we just do the same for the other solution family x = (-11k + SQRT(7)k)/6 = -1.392k, giving k = 0.718x. Substituting that back in to f(x) gives h(x) = (0.282x)(-0.436x)(-0.795x) = 0.09782757x3.
Last edited by Thommo on Apr 11, 2017 5:29 pm, edited 2 times in total.
User avatar
Thommo
 
Posts: 27174

Print view this post

Re: Question with maximum points of some curves...

#3  Postby newolder » Apr 11, 2017 5:13 pm

The y = x^3 curve is redundant and does not fit the 'maximum/minimum' criteria of the others - it has a point of inflexion at the origin. So the degree of the required polynomials could probably be reduced to 5, I guess. :dunno:
I am, somehow, less interested in the weight and convolutions of Einstein’s brain than in the near certainty that people of equal talent have lived and died in cotton fields and sweatshops. - Stephen J. Gould
User avatar
newolder
 
Name: Albert Ross
Posts: 7310
Age: 1
Male

Country: Feudal Estate number 9
Print view this post

Re: Question with maximum points of some curves...

#4  Postby LjSpike » Apr 12, 2017 11:51 am

Thommo wrote:
You've plotted a few examples of a family of curves, which have the function f(x)=(x+k)(x+2k)(x+2.5k).

Expanding that out: f(x) = x3 + 5.5kx2 + 9.5k2x + 5k3

Differentiate to find the turning points: f'(x) = 3x2 + 11kx + 9.5k2

Find the solutions by putting that into the quadratic formula with a = 3, b = 11k, c = 9.5k2 gives two solutions of x = (-11k + SQRT(7)k)/6 and (-11k - SQRT(7)k)/6 we can note that you've simply plotted the turning points stemming from the second solution by inspection.

This allows us to relate x and k by the formula x = (-11 - SQRT(7))k/6 or x = -2.274k (taking k to 3 d.p). Or equivalently k = -x/2.274.

Finally we can substitute that back into f(x) to get the curve of the solution family as g(x) = (0.560x)(0.121x)(-0.099x) = -0.006706415x3.

To answer the follow-on question we just do the same for the other solution family x = (-11k + SQRT(7)k)/6 = -1.392k, giving k = 0.718x. Substituting that back in to f(x) gives h(x) = (0.282x)(-0.436x)(-0.795x) = 0.09782757x3.


Thank you. The former bit to your post I had know, but the latter was helpful, and those equations do definitely fit the pattern. :P
LjSpike
THREAD STARTER
 
Posts: 89
Age: 20
Male

United Kingdom (uk)
Print view this post


Return to Mathematics

Who is online

Users viewing this topic: No registered users and 1 guest