What is the maximum value of the function f Rn R defined b

What is the maximum value of the function f : Rn ? R defined by f(x) = x21x2 ···x2n

onthesphereSn?1 ={x?Rn :xTx=1}?

2. What is the maximum value of the function f R\" Rdefined by flac) 2m2 ...zR on the sphere Sn-1 ERm z 1

Solution

Let ( x_1, x₂ , --- x_n) represent the point on the sphere

Given x₁²+ x₂²+-----x_n² =1 to find the maximum of the product [ x ₁² x₂² ---- x_n²]

this is the extension of the problem x+y =1 find the maximum of the product xy

Product = xy P(x) = x( 1-x) { as x+y=1 => y=1-x}

P \'(x) = 1- 2x =0 => x=1/2

P \'\'(x) = -2 hence the point x= 1/2 gives the maximum value of the product

when x=1/2 , y=1/2 { as x+y=1}

hence the maximum product of xy = 1/2 x1/2 = 1/4

Extendind to 3 variables x= y= z= 1/3 and the maximum product of xyz = 1/3 x1/3 x1/3 =1/27

extending to n variables x₁² = x₂² = ------- x_n²= 1/n

and the maximum product x₁² x₂² ----- x_n² = 1/n x1/n ------1/n ( n tmes )

= [1 / n]ⁿ