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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1 solution

Sanjoy Kundu
Jul 25, 2017

Let's look at some well known functions. When I think of a fast function, generally exponential functions come to mind so lets look at something that decreases very quickly - like exp(-x). A function that grows very slowly is log(x). So now using these two functions, lets see what we can tell about each of these statements

I. False. exp(-x)/log(x) is not defined at all reals (x = 1). Even though it is monotonically decreasing where it is defined, it is not monotonically decreasing throughout all reals.

II. False. The limit as x -> negative infinity does not exist, in our example. In fact, as x -> negative infinity, log(x) isn't even well defined!

III. False. While it is true that log(x) is monotonically increasing, this does not imply that it's derivative is strictly positive. Refer to the graph of 1/x.

IV. True. This is a well known statement about monotonic functions.

V. True. Another well known statement about monotonic functions. If you are not convinced, try creating a monotonic function with any other type of discontinuity, and see if it is possible.

VI. False. If the lim as x -> c yields 0/0, an indeterminate form, and exists, then f(x)/g(x) has a hole (removable discontinuity) at that point.

How well do you know functions?

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