Determine the greatest possible value of ∑i=110cos⁡3xi for real numbers x1,x2…x10 satisfying ∑i=010cos⁡xi=0 My attempt: ∑cos⁡3x=∑4cos3x−∑3cos⁡x=4∑cos3x 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.

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Determine the greatest possible value of  ∑i=110cos⁡3xi  for real numbers x1,x2…x10 satisfying  ∑i=010cos⁡xi=0  My attempt:   ∑cos⁡3x=∑4cos3x−∑3cos⁡x=4∑cos3x  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.

Ydaxq · Accepted Answer

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  −1≤x≤1   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 −m10−m

ul2ph3ojc · Accepted Answer

Hint: Use the fact that  cos⁡a+cos⁡b=2cos⁡(12a+b)cos⁡(12a−b)  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

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

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