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Function + tangent line = 0

  1. Aug 22, 2014 #1
    Say we have two functions with the following properties:

    [itex]f(x)[/itex] is negative and monotonically approaches zero as [itex]x[/itex] increases.
    [itex]g(x,y)[/itex] is a linear function in [itex]x[/itex] and is, for any given [itex]y[/itex], tangent to [itex]f(x)[/itex] at some point [itex]x_0(y)[/itex] that depends on the choice of [itex]y[/itex] in a known way.
    Additionally, for any given [itex]y[/itex], [itex]f(x) \leq g(x,y)[/itex] for all [itex]x[/itex], with equality only at [itex]x = x_0[/itex].

    It is then true that for each [itex]y[/itex], [itex]f(x) + g(x,y) = 0[/itex] at precisely one value of [itex]x[/itex]. I'm trying to find this value.

    Writing [itex]g[/itex] as a tangent line;

    [itex]g(x,y) = f(x_0(y)) + f'(x_0(y))(x - x_0(y))[/itex]

    seems to be the obvious place to start, but trying to solve the above equaiton I find myself forced to try to invert [itex]f(x)[/itex] at some point, which unfortunately cannot be done in terms of standard functions for the cases in which I'm interested.

    My question is thus; can this be done in a nice way? And if not generally, are there specific circumstances under which this may be done?

    Thank you!
     
  2. jcsd
  3. Aug 22, 2014 #2

    HallsofIvy

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    It is not necessary to solve that equation. If you show that, for some x, f(x)+ g(x,y) is negative, that for another x, f(x)+ g(x, y) is positive, and that f(x)+ g(x, y) is monotone, then there must exist a unique x such that f(x)+ g(x,y)= 0.
    (That's true if f(x) is continuous. You don't give that as a hypothesis but it is implied.)
     
  4. Aug 22, 2014 #3
    Thank you for your reply!

    Yes, f(x) is continuous. And indeed f(x) + g(x,y) is monotone.
    What I meant to ask was if there is a way to explicitly find that value for x at which f+g=0 other than the "brute force" way of inverting the expression? Or perhaps, more generally - does an equation of the form "function + tangent line = 0" have a known solution under some assumptions/circumstances?
     
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