We use cookies to personalise content and advertisements and to analyse access to our website. Furthermore, our partners for online advertising receive pseudonymised information about your use of our website. cookie policy and privacy policy.
 
+0  
 
+1
180
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+7761 
+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

15 Online Users

avatar