If `x,y,z` are positive real numbers such that `x^(2)+y^(2)+Z^(2)=7` and `xy+yz+xz=4` then the minimum value of `xy` is
If `x,y,z` are positive real numbers such that `x^(2)+y^(2)+Z^(2)=7` and `xy+yz+xz=4` then the minimum value of `xy` is
A. `1`
B. `(1)/(2)`
C. `(1)/(4)`
D. `(1)/(8)`
5 views
1 Answers
Correct Answer - C
`(c )` `xy=4-(x+y)z`
Now consider `x+y` and `z`
Using `G.M. ge A.M.`, we have
`sqrt((x+y)z) le (x+y+z)/(2)`
Also, `(x+y+z)^(2)`
`=x^(2)+y^(2)+z^(2)+2(xy+yz+zx)`
`=7+8=15`
Now `(x+y)z le ((x+y+z)^(2))/(4) le (15)/(4)`
`implies (x+y)z|_(max)=(15)/(4)`
`impliesxy|_(min)=4-(15)/(4)=(1)/(4)`
5 views
Answered