Minimum values of coefficients of a quadratic Given a function $f(x)= ax^{2} + bx + c$ where $a<b$ and $f(x)\geq{0}$ for all real values of x. Then how would one find the minimum value of the relation between coefficients of the give quadratic. For ex, 
How would one find the min value of $\frac{a+b+c}{b-a}$.
my work so far 
I concluded that $\frac{a+b+c}{b-a}$ is the same as $\frac{f(1)}{b-a}$ and as per the given conditions $b^2 -4ac\leq{0}$ and I tried finding some triplets of $a,b, c$ and find the minimum value by observation but had no luck. 
All help is greatly appreciated
 A: Let $f(x)=ax^2+bx+c$ where
$$\forall x \in \mathbb R, f(x) \ge 0 \qquad
\text{and} \qquad a < b \tag A$$
$$\text{Find $\min \dfrac{a+b+c}{b-a}$} \tag{B}$$
If $a=0$, then the problem is ill-defined. When $a < 0$, then $\displaystyle \lim_{x \to \infty}f(x) \to -\infty$. So $a > 0$.
Since $f(0) = c$, then we must have $ c \ge 0$, but we can do better than that. Clearly $f(x)$ is a parabola above or on the $x$-axis. So, holding $a$ and $b$ fixed, the value of $c$ that will minimize $\dfrac{a+b+c}{b-a}$ is the value, $c = -\frac 14b^2$, that puts the vertex of $f(x)$ on the $x$-axis. Combining that with $c \ge 0$, we get $c=0$. Since we require $a < b$, we let $b = aN$ where 
$N > 0$.
We get $f(x) = a(x^2 + Nx + 0)$ and we want to minimize 
\begin{align}
   \dfrac{a+b+c}{b-a} 
   &= \dfrac{a+aN+0}{aN-a} \\
   &= \dfrac{1+N}{N-1} \\
   &= 1 + \dfrac{2}{N-1}
\end{align}
which approaches $1$ as $N$ approaches infinity.
For example, $f(x) = x^2 + Nx + 0$, where $N > 1$ satisfies condition (A) and
$\dfrac{a+b+c}{b-a} =  \dfrac{N+1}{N-1} \to 1$ as $N \to \infty$.
A: I think there is a very simple graphical version of the solution given by Steven Gregory. Make the substitutions $b/a=B, c/a=C$. Then all of your information is about the two numbers $B,C$. For example they are both positive and $B^2<4C$. Show all of your information in the $(B,C)$ plane. The constraints will carve out a feasible region of this plane. The function that you wanted to minimise is $$(1+B+C)/(B-1)$$ whose contours in this plane are straight lines. The answers should be visually obvious, and can be backed up with algebra if you wish
This will only work if the functions you are interested in are homogeneous in $a,b,c$. but I am willing to wager that they are
