To solve the problem of finding the maximum value of ((a+b)(b+c)(c+a)) given positive real numbers (a, b, c) such that (a^2 + b^2 + c^2 = 1), follow these steps:

Key Observations and Symmetry

The expression ((a+b)(b+c)(c+a)) is symmetric in (a, b, c). For symmetric functions under constraints, the maximum often occurs at equal variables ((a=b=c)).

Case when (a=b=c)

If (a=b=c), then (3a^2 = 1 \implies a = \frac{1}{\sqrt{3}}).

Compute ((a+b)(b+c)(c+a)):
[ (2a)(2a)(2a) = 8a^3 ]

Substitute (a=\frac{1}{\sqrt{3}}):
[ 8\left(\frac{1}{\sqrt{3}}\right)^3 = \frac{8}{3\sqrt{3}} = \frac{8\sqrt{3}}{9} ]

Verification with Lagrange Multipliers

Using Lagrange multipliers confirms that the critical point is at (a=b=c), so this is indeed the maximum.

Answer: (\boxed{\dfrac{8\sqrt{3}}{9}})

(\boxed{\dfrac{8\sqrt{3}}{9}})

作者声明:本文包含人工智能生成内容。