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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