# How well do you know functions?

Calculus Level 5

There exists a monotonically decreasing function $$f(x)$$ and a monotonically increasing function $$g(x)$$ for $$x$$ spanning the positive reals. After graphing the two functions, I realize that $$f(x)$$ is decreasing at a faster rate than $$g(x)$$ is increasing. I claim six statements about the functions. Determine how many of them are true, given that at least two of them are true.

I. $$\dfrac {f(x)}{g(x)}$$ is a monotonically decreasing function, for all $$x$$ spanning the reals.

II. As $$x$$ approaches positive or negative infinity, the limit of $$\dfrac {f(x)}{g(x)}$$ exists.

III. The first derivatives of $$f(x)$$ and $$g(x)$$ are strictly negative and positive respectively, as $$x$$ spans the reals.

IV. $$f(x)$$ and $$g(x)$$ can only have a countably finite number of discontinuities.

V. Any discontinuities present in $$f(x)$$ and $$g(x)$$ must be jump discontinuities.

VI. Any discontinuities present in $$\dfrac {f(x)}{g(x)}$$ must be jump discontinuities.

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