If a,b,c are positive real numbers and
$z = \frac{b^2 + c^2}{b+c} + \frac{c^2 + a^2}{a+c} + \frac{a^2+b^2}{a+b}$
then only one of the following statements is always true , which on is it ?
a) $0<=z<a$
b.) $a<=z<a+b$
c.)$a+b<=z<a+b+c$
d.) $a+b+C <= z <2(a+b+c)$
My attempt : I tried adding the different terms up and then separating by completing the squares ( such as turning $b^+c^2$ into $(b+c)^2$ by adding and subtracting 2bc) However couldn't get much far.
I think I need to figure out a way to apply the A.M.-G.M. inequality here , but couldn't find a proper way of doing so.
Any help will be appreciated !
D.
To see the upper bound, your expression is less than obtained by adding $2bc$ in numerator of first, $2ac$ in second and $2ab$ in third where you get that this is $\le a+b+a+c+b+c = 2(a+b+c)$.
For lower bound, we need to show that $$\frac{bc}{b+c} + \frac{ab}{a+b}+\frac{ac}{a+c}\le \frac{a+b+c}{2}$$ This can be shown since $$\frac{bc}{b+c} \le \frac{b+c}{4}$$ and terms can be added up.