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Question

Let a, b, c, and d be positive real numbers such that a + b + c + d = 10. Find the maximum value of ab^2 c^3 d^4.

Accepted Answer

By AM-GM, align* a + b + c + d &= a + b2 + b2 + c3 + c3 + c3 + d4 + d4 + d4 + d4 & 10 [10]a ( b2 )^2 ( c3 )^3 ( d4 )^4 &= 10 [10]ab^2 c^3 d^427648. align*Since a + b + c + d = 10, ab^2 c^3 d^4 27648.Equality occurs when a = 1, b = 2, c = 3, and d = 4, so the maximum value is 27648.