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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

Saakshya Devat
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  • Basically yes I wish to know how to minimize a function made up of the coefficients of a quadratic given that the quadratic has some conditions imposed on it and the least straightforward example I could think of was a fractional relation. This particular example only serves to ask a guideline of sorts. As if one can work with division of the variables involved it usually possible to work with other elementary operators such as addition subtraction and multiplication. – Saakshya Devat Apr 12 '17 at 13:55
  • And maybe I have chosen a tag which is not appropriate due to my inexperience and if that is true then I deeply apologize and request you to tell me what category does this query actually fall in. – Saakshya Devat Apr 12 '17 at 13:56
  • Along what lines do you think it is not well defined? It wasn't actually a question but a query that was created due to my own curiosity. – Saakshya Devat Apr 13 '17 at 01:53
  • I agree that additional constraints might remove some ambiguities currently present and also the main reason for your trail of thought which I to a great extent agree with is probably due to the fact that it is not a full fledged problem, it wasn't created by mathematicians it is a result of my curiosity on something I was at the time dealing with. And yes I'll change the tag, hopefully more people see this and more views can be considered by you me and Steven Gregory. – Saakshya Devat Apr 13 '17 at 02:58
  • I hope it's not too much to ask but I've added another questions ( it is not a query but an genuine full fledged genuine question) if you get the time would you mind taking a look at it. – Saakshya Devat Apr 13 '17 at 03:01

2 Answers2

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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 $N \to \infty$.

Steven Alexis Gregory
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    Thank you for your contribution Steven. Greatly appreciate it. – Saakshya Devat Apr 13 '17 at 03:03
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    Oh, is that so? I'll also try and work upon what you've wrote so far and if I come across something interesting I'll definitely let you and nick know – Saakshya Devat Apr 13 '17 at 06:57
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    @Nick but then again the minimum value is asked so the assumption is valid, isn't it? – Saakshya Devat Apr 14 '17 at 01:44
  • I don't know what you mean by "valid": it's your problem and you can add whatever assumptions you want. "Valid" implies some outside control (e.g. if the problem arose out of a physical situation, the assumption might be valid or invalid for that situation - but AFAICT, there is no such outside control so there is no validity or invalidity associated). It does make @Steven Gregory's solution complete AFAICT. Steven Gregory deleted all his comments I guess which made the discussion very one-sided :-) so I went ahead and deleted all my comments too... – NickD Apr 14 '17 at 14:36
  • .. unfortunately leaving some dangling pointers, but that can't be helped. – NickD Apr 14 '17 at 14:36
  • @nick - MSE tries to discourage "chatting" in the comment section. I've gotten in trouble before for that. – Steven Alexis Gregory Apr 14 '17 at 17:51
  • @Nick - I went back and looked at what did and you were right and I was wrong. I have changed my answer and I hope that it meets with your approval. – Steven Alexis Gregory Oct 16 '17 at 02:04
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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

Philip Roe
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