Home
Class 12
MATHS
If a, b, c are positive real numbers suc...

If `a, b, c` are positive real numbers such that `a + b + c = 1`, then prove that `a/(b + c)+b/(c+a) + c/(a+b) >= 3/2`

Text Solution

Verified by Experts

Arithmetic mean of `(-1)`th powers `le (-1)`th power of arithmetic mean
`((((b+c)/(a+b+c))^(-1)+((c+a)/(a+b+c))^(-1)+((a+b)/(a+b+c))^(-1))/(3))ge (((b+c)/(a+b+c)+(c+a)/(a+b+c)+(a+b)/(a+b+c))/(3))^(-1)`
`implies ((a+b+c)/(b+c)+(a+b+c)/(c+a)+(a+b+c)/(a+b))/(3)ge ((2)/(3))^(-1)`
`implies (a)/(b+c)+1+(b)/(c+a)+1+(c)/(a+b)+1 ge (9)/(2)`
`implies (a)/(b+c)+(b)/(c+a)+(c)/(a+b) ge (9)/(2)-3`
or ` (a)/(b+c)+(b)/(c+a)+(c)/(a+b) ge (3)/(2)`.
Promotional Banner

Similar Questions

Explore conceptually related problems

If a,b,c,d are positive real numbers such that a+b+c +d = 12 , then M = ( a+ b ) ( c + d ) satisfies the relation :

If a, b and c real numbers such that a^(2) + b^(2) + c^(2) = 1, then ab + bc + ca lies in the interval :

If a, b, c are distinct positive real numbers and a^(2)+b^(2)+c^(2)=1 , then ab+bc+ca is :

If a,b,c and d are four positive real numbers such that abcd=1 , what is the minimum value of (1+a)(1+b)(1+c)(1+d) .

If a, b, c are positive real numbers, then the number of real roots of the equation a x^(2)+b|x|+c=0 is

If a,b,c are positive real numbers, then the roots of the equation ax^(2) + bx + c =0

If three positive real numbers a,b,c are in A.P. such that abc=4, then the minimum value of b is

If a,b,c are three unequal numbers such that a,b,c are in A.P. And b-a, c -b , a are in G.P., then a: b : c is :

If a,b,c are in HP, then prove that (a+b)/(2a-b)+(c+b)/(2c-b)gt4 .

If a,b, and c are positive integers such that a+b+c le8 , the number of possible values of the ordered triplet (a,b,c) is