Determine the greatest possible value of \sum_{i=1}^{10} \cos 3x_i for real numbers PS

Adrien Barron 2022-01-23 Answered
Determine the greatest possible value of
i=110cos3xi
for real numbers x1,x2x10 satisfying
i=010cosxi=0
My attempt:
cos3x=4cos3x3cosx=4cos3x
So now we have to maximize sum of cubes of ten numbers when their sum is zero and each lie in interval [−1,1]. i often use AM GM inequalities but here are 10 numbers and they are not even positive. Need help to how to visualize and approach these kinds of questions.
You can still ask an expert for help

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more

Solve your problem for the price of one coffee

  • Available 24/7
  • Math expert for every subject
  • Pay only if we can solve it
Ask Question

Expert Answer

Ydaxq
Answered 2022-01-24 Author has 12 answers
Visualising the solution
You have asked for help in visualising the solution. I think you will find it useful to have in mind the picture of y=x3  for  1x1
Now consider the arrangement of the 10 numbers in the maximum position. (We have a continuous function on a compact set and so the maximum is attained.)
First suppose that there is a number, s, smaller in magnitude than the least negative number l. Increasing l whilst decreasing s by the same amount would increase the sum of cubes and therefore cannot occur.
So, all the negative numbers are equal, to l say, and all the positive numbers are greater than |l|.
Now suppose that a positive number was not 1. Then increasing it to 1 whilst reducing one of the ls would increase the sum of cubes and therefore cannot occur.
Hence we need only consider the case where we have m 1s and 10−m numbers equal to m10m
Not exactly what you’re looking for?
Ask My Question
ul2ph3ojc
Answered 2022-01-25 Author has 12 answers
Hint: Use the fact that
cosa+cosb=2cos(12a+b)cos(12ab)
If you pair the summands and apply above transformation, then the sum becomes a product with 10 cosine factors, and a scaling factor of 25. So now at least we have an estimate of the sum
Not exactly what you’re looking for?
Ask My Question

Expert Community at Your Service

  • Live experts 24/7
  • Questions are typically answered in as fast as 30 minutes
  • Personalized clear answers
Learn more