quadratic equations intersection point is minimum instead of roots!

[Gevni]

Thanks JeffM for the detailed reply. Little bit background what I did so far to solve my problem.
(1) First I drives the 2 formulas that will calculate the cost of my problem for two different scenarios and I have to choose the one which has minimum cost.
(2) Next I find the range for both formulas that is "t" which define the range for both scenarios like x<=t for f(x) and x>=t for g(x).
(3) Took first derivative of these formula separately f(x), and g(x).
(4) Equal these equations to zero
(5) Rearrange them to become quadratic equations
(6) Using quadratic formula solve them for the roots of f(x) and g(x)
(7) if value of "a that is (A-B) for f(x) and (C+D) for g(x)" is negative then ignore that case as roots are giving the maximum value.
(8) Else if "a" for any or both functions is positive and roots of f(x) and or g(x) are real and in range of that function, I consider roots and "t" as a candidates that will give me minimum value for that function. evaluate f(root1), f(root2) and f(t), which evaluation gives me lowest cost is my answer for f(x) and same I did for g(x).

I got the results, whatever value I gave to A,B C,D the result shows f(t) will be the optimal solution. So I was thinking to write a proof that shows that between range of 0 <x <n and two function f(x) anf g(x) in two above intervals, f(t) will always be an optimal minimum point. Mean f(x) decrease and g(x) increase at point f(t) And here I am stuck, how to write this in proof. I don't know if it make sense to you now.

Now we are getting somewhere.

 

[JeffM]

Thanks JeffM for the detailed reply. Little bit background what I did so far to solve my problem.
(1) First I drives the 2 formulas that will calculate the cost of my problem for two different scenarios and I have to choose the one which has minimum cost.
(2) Next I find the range for both formulas that is "t" which define the range for both scenarios like x<=t for f(x) and x>=t for g(x).
(3) Took first derivative of these formula separately f(x), and g(x).
(4) Equal these equations to zero
(5) Rearrange them to become quadratic equations
(6) Using quadratic formula solve them for the roots of f(x) and g(x)
(7) if value of "a that is (A-B) for f(x) and (C+D) for g(x)" is negative then ignore that case as roots are giving the maximum value.
(8) Else if "a" for any or both functions is positive and roots of f(x) and or g(x) are real and in range of that function, I consider roots and "t" as a candidates that will give me minimum value for that function. evaluate f(root1), f(root2) and f(t), which evaluation gives me lowest cost is my answer for f(x) and same I did for g(x).

I got the results, whatever value I gave to A,B C,D the result shows f(t) will be the optimal solution. So I was thinking to write a proof that shows that between range of 0 <x <n and two function f(x) anf g(x) in two above intervals, f(t) will always be an optimal minimum point. Mean f(x) decrease and g(x) increase at point f(t) And here I am stuck, how to write this in proof. I don't know if it make sense to you now.

If you gave me the correct equations for f(x) and g(x), you should NOT have got quadratics for the first and second derivatives. The first derivative of a quadratic is a linear function, and the quadratic formula has nothing to do with linear functions. Moreover, the roots of a function and the roots of its derivative coincide only in special cases. So steps 5 and 6 appear to be incorrect in general though they may work in special cases..

I am guessing that you worked on these by substituting numbers for a, b, c, d, n, and t. Post those values and your results for each step, and tomorrow we can go from there.

 

[Gevni]

Just to clarify, I took the 1st derivative of my original cost formulas f(x) anf g(x) which results into quadratic equations. I didn't took derivative of quadratic equation.

 

[JeffM]

Just to clarify, I took the 1st derivative of my original cost formulas f(x) anf g(x) which results into quadratic equations. I didn't took derivative of quadratic equation.

Please tomorrow give the cost functions themselves. In post 19, you specified that f(x) and g(x) were quadratics, and you said "the cost of my formula f(x) or g(x)." Why would I presume that f(x) and g(x) were not cost functions? On what basis was I to conclude that they are marginal cost functions?

We have wasted a huge amount of time by starting without explicitly identifying that you are trying to solve a problem involving alternate cost functions that seem to apply to different ranges of product, without actually specifying the two cost functions, and without sharing any information where, if anywhere, the domains of the two functions may overlap or what the quantitative relationship between the domains is.

No one can say whether you have made an error somewhere or whether you are firing on all cylinders. We cannot compute roots, extrema, or intersections without formulas. Nor can we provide an answer without fully understanding the constraints. You say x can be zero, but that usually implies no revenue so maybe you mean x must be greater than zero. And I have no idea how, if at all, n and t are related. We are not magicians. We need all relevant information expressed as clearly and completely as possible. You would never say to a lawyer, "Oh, no why did you think the contract I showed you is one I am being sued over," or to a doctor, "Oh no, did I forget to tell you about my heart surgery." If we charged billable hours, I suspect that we would had not have spent most of a day just collecting information you could have provided in your initial post. (My billing rate is 400 dollars per hour. If I were charging at this site, I'd be billing you for over 2,000 US dollars. Fortunately for you and unfortunately for me, this is a free site.)

 

[Gevni]

Sorry JeffM for wasting your time. I tried to simplify the problem that turn into a mess. Any how, I am closing my thread so you don't feel unfortunate. I better spend some time on background reading and eventually will come to the solution.
Many thanks for your time.
PS: I never said x=0, I always wrote 0<x<n