+0  
 
+1
194
2
avatar

Let \(x_1,x_2,\dots, x_{101}\) be positive real numbers such that \(x_1^2 + x_2^2 + \dots + x_{101}^2 = 1.\) Find the maximum value of \(x_1 x_2 + x_1 x_3 + \dots + x_1 x_{101}.\)

 

 Mar 8, 2019
 #1
avatar+7763 
+1

Maximum occurs at \(x_1 = x_2 = x_3 = \cdots = x_{101} = \dfrac{1}{\sqrt{101}}\).

Maximum value is \(100\left(\dfrac{1}{\sqrt{101}}\right)^2 =\dfrac{100}{101}\).

 

Notes: The equation \(x_1^2 + x_2^2 + \cdots + x_{101}^2 = 1\) represents a 101-sphere with radius 1.

 

:)

 Mar 9, 2019
 #2
avatar+6046 
+1

I believe your answer but prove it.

Rom  Mar 9, 2019

14 Online Users

avatar