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5.8 Show that the maximum value of the function f(x) = x₁²x₂²x² on the sphere Sn-1 = {XER" : x = 1} is (1/n)". That is (x₁2x2) ¹/1/n if x = S-¹.

5.8 Show that the maximum value of the function f(x) = x₁²x₂²x² on the sphere Sn-1 = {XER" : x = 1} is (1/n)". That is (x₁2x2) ¹/1/n if x = S-¹.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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5.8 Show that the maximum value of the function f(x)= x₁²x2²x² on the sphere Sn-1={XER" x = 1} is (1/n)". That is (x₁2x2) ¹/1/n if x = S"-1. Given n positive numbers a₁, ..., an, define a; ¹/2 . X₁ = Then x₁² + ... + x² = 1, so an + an)") 1/n (α₁ + 1 +an) ¹/2' n a₁ (a₁ + Thus the geometric mean of n positive numbers is no greater than their arithmetic mean. i = 1,..., n. or 1 (a,... an) ¹/n n (a₁ +...+an).
Transcribed Image Text:5.8 Show that the maximum value of the function f(x)= x₁²x2²x² on the sphere Sn-1={XER" x = 1} is (1/n)". That is (x₁2x2) ¹/1/n if x = S"-1. Given n positive numbers a₁, ..., an, define a; ¹/2 . X₁ = Then x₁² + ... + x² = 1, so an + an)") 1/n (α₁ + 1 +an) ¹/2' n a₁ (a₁ + Thus the geometric mean of n positive numbers is no greater than their arithmetic mean. i = 1,..., n. or 1 (a,... an) ¹/n n (a₁ +...+an).
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