Algebra

Power Mean Inequalities

AM-GM

         

Suppose the average of 5 5 positive numbers is equal to 2. 2. What is the maximum possible product of the numbers?

Suppose xyz=8 xyz = 8 with x,y,z>0.x, y, z > 0. What is the minimum value of x+y+z x + y + z ?

What is the largest n n for which

x3+y3+z3nxyz x^3 + y^3 + z^3 \geq nxyz

holds for all x,y,z>0 x, y, z > 0 ?

What is the largest n n for which

(x+y)(y+z)(x+z)nxyz (x + y)(y + z)(x + z) \geq nxyz

holds for all positive x,y,z x, y, z ?

Suppose 2x+3y+z=18. 2x + 3y + z = 18. Find the maximum value of xyz xyz for x,y,z>0. x, y, z > 0.

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