SELECTION EXAMINATION

Question

If x, y, z are positive real numbers, prove that: (3x + y)(3y + z)(3z + x) 64xyz. When does equality hold?

Accepted Answer

We use the inequality of arithmetic and geometric means for four positive terms:

\frac{a _{1} + a _{2} + a _{3} + a _{4}}{4} \geq 4 a_{1} a_{2} a_{3} a_{4},

where equality holds when

a_{1} = a_{2} = a_{3} = a_{4}

.

Applying this to each factor:

(3 x + y) \geq 4 4 x^{3} y

(3 y + z) \geq 4 4 y^{3} z

(3 z + x) \geq 4 4 z^{3} x

Multiplying these inequalities:

(3 x + y) (3 y + z) (3 z + x) \geq 4 4 x^{3} y \cdot 4 4 y^{3} z \cdot 4 4 z^{3} x = 64 4 x^{4} y^{4} z^{4} = 64 x y z .

Equality holds when

x = y = z

.