SOLUTION: Let x, a, b, c be nonnegative real numbers. Prove that (x + cuberoot{abc})^3 <= (x + a)(x + b)(x + c) <= ( x + {(a+b+c)/(3)}^3)

Algebra ->  Inequalities -> SOLUTION: Let x, a, b, c be nonnegative real numbers. Prove that (x + cuberoot{abc})^3 <= (x + a)(x + b)(x + c) <= ( x + {(a+b+c)/(3)}^3)      Log On


   

Question 1179366: Let x, a, b, c be nonnegative real numbers. Prove that
(x + cuberoot{abc})^3 <= (x + a)(x + b)(x + c) <= ( x + {(a+b+c)/(3)}^3)
Answer by math_helper(2461) About Me  (Show Source):
You can put this solution on YOUR website!



See this:

https://en.wikipedia.org/wiki/Inequality_of_arithmetic_and_geometric_means



and



https://math.stackexchange.com/questions/1298115/prove-inequality-abbcca-ge-3-abc-1






If you expand the left hand side you will get:



     = 



Expanding the middle gives:

    +%28x%2Ba%29%28x%2Bb%29%28x%2Bc%29 = x%5E3+%2B+%28a%2Bb%2Bc%29x%5E2+%2B+%28ab%2Bac%2Bbc%29x+%2B+abc+



Comparing term-by term, you get two inequalities to prove:

     

     



The LHS is a GM, the RHS is an AM,  so here the AM-GM inequality applies.



The middle term compared to the RHS (in original problem statement) is then proven in a similar fashion.