... found¹

This method can be justified by plugging back into the original equation and verifying that the result is a solution.

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...typically²

It is not necessary that $\lim_{x\rightarrow\pm\infty} f(x) = 0$ for the Fourier transform to exist but it is satisfied in most every case. The condition that the Fourier transform exists is that

$$\int_{-\infty}^{\infty} \vert f(x)\vert dx$$

exists and is bounded.

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...?³

An amusing variation on this problem would be to find the path that the path that a winning downhill skier should traverse.

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