Maximum

oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?

1)100
2)100/3
3)10/3
4)3

Did you make up that question, or does it come from a particular source?

It's not a standardized GMAT test but I saw it in a compiled source for the Algebraic section of GMAT.the level of the test is high and in my country (Iran) GMAT questions present in 4-option form.

oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?

1)100
2)100/3
3)10/3
4)3

should be 10/3

we know that AM >= GM
so if we consider the terms of the series as 5xy,9yz & 2xz and apply the above principle
(5xy+9yz+2xz)/3 >= (5xy.9yz.2xz)^(1/3)
=> (5xy+9yz+2xz)/3 >= (90.x^2.y^2.z^2)^(1/3) - (1)
but we know
5xy+9yz+2xz=30 from given question
substitute in equation (1)
30/3 > = (90.x^2.y^2.z^2)^(1/3)
10 >= (90.x^2.y^2.z^2)^(1/3)
cubing on both sides
1000 >= 90.x^2.y^2.z^2
=>xyz <= 10/3
=> max of xyz is 10/3
hope this helps

user123321

we know that AM >= GM

I don't understand what exactly it means!

it's obvious, y and z should be minimized and x maximized -> y=z=1 and 5x+9+2x=30, 7x=21, x=3
and xyz=3
4

oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?

1)100
2)100/3
3)10/3
4)3

pemdas wrote:it's obvious, y and z should be minimized and x maximized

why is that? why do u minimize exactly y and z?because of stated answer choices?

imho, the q. is not a gmat q.

no, because we are dealing with polynomials and I'm assessing the coefficients of the given polynomials. By minimizing the highest coefficient polynomials I am increasing the value of xyz.

LalaB wrote:

pemdas wrote:it's obvious, y and z should be minimized and x maximized

why is that? why do u minimize exactly y and z?because of stated answer choices?

imho, the q. is not a gmat q.

pemdas wrote:it's obvious, y and z should be minimized and x maximized -> y=z=1 and 5x+9+2x=30, 7x=21, x=3
and xyz=3
4

oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?

1)100
2)100/3
3)10/3
4)3

Pemdas, the assumptions that y=z=1 are valid only if we are told that x,y,z are integers > 0.
otherwise, they can assume much smaller values...

oldheaven wrote:

we know that AM >= GM

I don't understand what exactly it means!

Take any set of non -ve numbers,
There is a proven theorem which says that always
Arthmetic Mean(AM) will be greater than or equal to Geometric Mean(GM).

In mathematics, the inequality of arithmetic and geometric means, or more briefly the AM–GM inequality, states that the arithmetic mean of a list of non-negative real numbers is greater than or equal to the geometric mean of the same list; and further, that the two means are equal if and only if every number in the list is the same.

src: https://en.wikipedia.org/wiki/Inequality ... tric_means
you can try with few numbers.

thank you user123321, I really appreciate your method and approach.

oldheaven, are you sure we should be preped for GM included as concept with this q.?
i barely remember this concept, not to speak about its application for test

tomada, i totally agree with you here

tomada wrote: Pemdas, the assumptions that y=z=1 are valid only if we are told that x,y,z are integers > 0.
otherwise, they can assume much smaller values...

see user also assumed non -ve values, the condition for GM as product of the substitutable is taken under root which may be even ...

again, i would mark xyz=3 if sitting for GMAT and using only polynomials background
thanks to oldheaven and user we remembered now some math/stats concepts