# Making a function $f(n)$ tangential to another function $g(n)$ by choosing appropriate constants

I have two functions $$f(n) = 1/(1+n) + b, (n \geq 0,b \geq 0)$$ and $$g(n) = pn^2 + qn + r$$. Assuming that I know the value of $$p, q$$, and $$r$$, I am trying to find a value for the constant $$b$$ such that $$f(n)$$ and $$g(n)$$ intersect at most one point while keeping the shortest distance between the functions as small as possible. (I am trying to find a tight upper bound on a parabola in the first quadrant.)

Currently, I am starting with a large value of $$b$$ and iteratively decreasing it by a small amount until Solve[f(n) == g(n), {n}] returns one or zero solutions. Can you please suggest an efficient way to do this?

Please find the code below.

ClearAll["Global*"]
testFun = 1/(1 + Subscript[n, 1]) + b;
g = 2.23 - 0.32*Subscript[n, 1] - 0.2*Subscript[n, 1]^2
For[bNum = 10, bNum >= 0, bNum -= 0.2,
numSol = Length[Solve[(testFun /. {b -> bNum}) == g && Subscript[n, 1] >= 0, {Subscript[n, 1]}]];
p = Plot[{testFun /. {b -> bNum}, g}, {Subscript[n, 1], 0, 100}, PlotRange -> {0, 15}];
Print[numSol];
Print[p];
If[numSol >= 1, Break[]]];
bNum = If[numSol == 1, bNum, bNum + 0.2]
`
• Code that can be copied and run often helps attract users who could help you. – Michael E2 Dec 9 '18 at 0:39
• @MichaelE2, thank you for the suggestion. I have added the code. – gaganso Dec 9 '18 at 1:02