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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