Say we have two functions with the following properties:(adsbygoogle = window.adsbygoogle || []).push({});

[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!

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# Function + tangent line = 0

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