Home
Class 12
MATHS
If x,y,z are positive real numbers such ...

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)`

Text Solution

Verified by Experts

The correct Answer is:
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)`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Comprehension|2 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE|Exercise Examples|37 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE|Exercise Solved Examples And Exercises|222 Videos

Similar Questions

Explore conceptually related problems

The x , y , z are positive real numbers such that (log)_(2x)z=3,(log)_(5y)z=6,a n d(log)_(x y)z=2/3, then the value of (1/(2z)) is ............

Given that x,y,z are positive real numbers such that xyz=32 , the minimum value of sqrt((x+2y)^(2)+2z^(2)-15) is equal to

If x,y,z be three positive numbers such that xyz^(2) has the greatest value (1)/(64) , then the value of (1)/(x)+(1)/(y)+(1)/(z) is

If g(x)=max(y^(2)-xy)(0le yle1) , then the minimum value of g(x) (for real x) is

Given that x ,y ,z are positive real such that x y z=32. If the minimum value of x^2+4x y+4y^2+2z^2 is equal m , then the value of m//16 is.

Given that x,y,z are positive reals such that xyz=32 . The minimum value of x^2+4xy+4y^2+2z^2 is ___________.

If x ,y ,a n dz are positive real umbers and x=(12-y z)/(y+z) . The maximum value of x y z equals.

If ( x +y+z ) = 9 and ( xy + yz + zx ) = 26 then find the value of x^(2) + y^(2) z^(2)