Let $a,$ $b,$ and $c$ be positive real numbers.  Find the set of all possible values of
\[\frac{c}{a} + \frac{a}{b + c} + \frac{b}{c}.\]
So we want to find the minimum value of the expression: $\frac{c}{a} + \frac{a}{b + c} + \frac{b}{c}$. Ok, so we want to minimize $\frac{c}{a} + \frac{a}{b+c}$. Right, so let's just write $L(a) = \frac{c}{a} + \frac{a}{b+c}$ and find its minimum. The derivative of $L$ is $\frac{-c}{a^2} + \frac{1}{b+c}$. So we want to solve $\frac{-c}{a^2} + \frac{1}{b+c} = 0$. So we can rewrite the expression as $\frac{c}{\sqrt{c(b+c)}} + \frac{\sqrt{c(b+c)}}{b+c} + \frac{b}{c}$. So we want to minimize $\frac{2\sqrt{c(b+c)}}{b+c} + \frac{b}{c}$. Right.  Let's denote this function by $M(b)$. So we want to minimize $M(b) = \frac{2\sqrt{c(b+c)}}{b+c} + \frac{b}{c}$. The derivative of $M$ is $-1(b+1)^{-1.5} + 1$. So we want to solve $-1(b+1)^{-1.5} + 1 = 0$. This is equivalent to $(b+1)^{1.5} = 1$. So $b+1=1$ which means $b=0$. But $b$ is positive. So the set of possible values is $(2,\infty)$.

