Home
Class 12
MATHS
If x,y,z gt 0 and x + y + z = 1, the pr...

If x,y,z `gt` 0 and x + y + z = 1, the prove that
`(2x)/(1 - x) + (2y)/(1 - y) + (2z)/(1 - z) ge 3`.

Text Solution

Verified by Experts

Using `A.M. ge H.M.` we have
`((x+y)+(y+z)+(z+x))/(3) ge (3)/((1)/(x+y)+(1)/(y+z)(1)/(z+x))`
` rArr 2(x+y+z)((1)/(x+y)+(1)/(y+z)+(1)/(z+x))ge 9`
`rArr 2((x+y+z)/(x+y)+(x+y+z)/(y+z)+(x+y+z)/(z+x))ge 9`
`rArr 2[(1+(z)/(x+y))+(1+(x)/(y+z))+(1+(y)/(x+z))]ge 9`
` rArr 2((x)/(y+z)+(y)/(z+x)+(z)/(x+y))ge3`
`rArr 2((x)/(1-x)+(y)/(1-y)+(z)/(1-z)) ge3`
Promotional Banner

Topper's Solved these Questions

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.3|6 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.4|4 Videos

  • INEQUALITIES INVOLVING MEANS

    CENGAGE PUBLICATION|Exercise Concept Application Eexercises 6.1|8 Videos

  • INEQUALITIES AND MODULUS

    CENGAGE PUBLICATION|Exercise Single correct Answer|21 Videos

  • INTEGRALS

    CENGAGE PUBLICATION|Exercise All Questions|762 Videos

Similar Questions

Explore conceptually related problems

If x > 0, y >0, z > 0 and x +y + z = 1 , prove that (1+x)(1+y)(1+z)ge 8(1-x) (1-y) (1-z)

If x+z = 2y " and " b^2 = ac , then prove that a^(y-z)*b^(z-x)*c^(x-y) =1 .

If y is the H.M. between x and z, prove that, (1)/(y-x) + (1)/(y-z) = (1)/(x) + (1)/(z) .

If z=x+iy and |2z-1| =|z-2|then prove that x^2+y^2=1

If x+y+z=xyz then prove that (3x-x^3)/(1-3x^2)+(3y-y^3)/(1-3y^2)+(3z-z^3)/(1-3z^2)=(3x-x^3)/(1-3x^2).(3y-y^3)/(1-3y^2).(3z-z^3)/(1-3z^2)

If (111.1)^x=(11.11)^y=(1.111)^z then prove that (1)/(x)-(2)/(y)+(1)/(z)=0 .

If x+y+z=x y z prove that (2x)/(1-x^2)+(2y)/(1-y^2)+(2z)/(1-z^2)=(2x)/(1-x^2)(2y)/(1-y^2)(2z)/(1-z^2)dot

If x + y + z = xyz and x, y, z gt 0 , then find the value of tan^(-1) x + tan^(-1) y + tan^(-1) z

If z=x+iy and |2z+1|=|z-2i|,then prove that 3(x^2+y^2)+4(x+y)=3

If z=x+iy and |2z-1|=|z-2|," prove that" " " x^(2)+y^(2)=1.