Did you make up that question, or does it come from a particular source?oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?
1)100
2)100/3
3)10/3
4)3
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tomada
- Master | Next Rank: 500 Posts
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I'm really old, but I'll never be too old to become more educated.
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.
should be 10/3oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?
1)100
2)100/3
3)10/3
4)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
Just started my preparation 
Want to do it right the first time.
Want to do it right the first time.
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
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
Success doesn't come overnight!
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:why is that? why do u minimize exactly y and z?because of stated answer choices?pemdas wrote:it's obvious, y and z should be minimized and x maximized
imho, the q. is not a gmat q.
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tomada
- Master | Next Rank: 500 Posts
- Posts: 406
- Location: Syracuse, NY
- Thanked: 23 times
- GMAT Score:740
Pemdas, the assumptions that y=z=1 are valid only if we are told that x,y,z are integers > 0.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
4oldheaven wrote:If 5xy+9yz+2xz=30 , then what is the maximum of xyz ?
1)100
2)100/3
3)10/3
4)3
otherwise, they can assume much smaller values...
I'm really old, but I'll never be too old to become more educated.
Take any set of non -ve numbers,oldheaven wrote:I don't understand what exactly it means!we know that AM >= GM
There is a proven theorem which says that always
Arthmetic Mean(AM) will be greater than or equal to Geometric Mean(GM).
src: https://en.wikipedia.org/wiki/Inequality ... tric_meansIn 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.
you can try with few numbers.
Just started my preparation
Want to do it right the first time.
Want to do it right the first time.
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
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
i barely remember this concept, not to speak about its application for test
tomada, i totally agree with you here
see user also assumed non -ve values, the condition for GM as product of the substitutable is taken under root which may be even ...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...
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
Success doesn't come overnight!
