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Simple idea, but I cannot figure out a way to do this. What I want is to find for which value of $k$ some equation $f(x,k)=g(x)$ has exactly one real solution.

For example, say $f(x,k)=x^2+k$ and $g(x)=6x^2+3$. Looking at this particular example, it's not too hard to figure out the value of $k$ we want is $3$. However, it's much more difficult for more complex functions, like $f(x,k)=kx^4-x^3$ and $g(x)=\ln(x)-x^2$. Looking at the graphs of these functions, it's easy to see that there must be exactly one value of $k$ where they are equal exactly once. Finding said value, however, eludes me.

The nature of this problem leaves it begging to be solved using limits, and calculus is never too far behind when limits are involved, but I'm not too familiar with multivariable calculus...

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closed as too broad by Luca Bressan, Dietrich Burde, Matthew Daly, N. F. Taussig, Lee David Chung Lin Sep 12 at 1:52

Please edit the question to limit it to a specific problem with enough detail to identify an adequate answer. Avoid asking multiple distinct questions at once. See the How to Ask page for help clarifying this question. If this question can be reworded to fit the rules in the help center, please edit the question.

  • $\begingroup$ The answer strongly depends on the forms of $f$ and $g$ $\endgroup$ – caverac Sep 11 at 18:16
  • $\begingroup$ Some equation? This is too broad. Most equations will not work, e.g., $f(x,k)=(x^2+1)^k$. $\endgroup$ – Dietrich Burde Sep 11 at 18:42

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