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Let S be the set of all infinitely differentiable functions whose domain and range are the real number. Let $S_1, S_2, S_3, $ and $S_4$ be the fours subsets of the S defined below:

$S_1$ = The set of infinity differentiable functions $f(x)$ such that $ \int_{0}^{3} f(x) dx = 0 $

$S_2$ = The set of infinity differentiable functions $f(x)$ such that $ f''(x) + 2\ln(x)f'(x) = \sin(x)f(x) $

$S_3$ = The set of infinity differentiable functions $f(x)$ such that $f(x)$ is either always increasing, always decreasing, or $f(x) = 0 $

$S_4$ = The set of infinity differentiable functions $f(x)$ such that $ (f''(x))^2 = 2f(x)$

How many of the above subsets satisfy both of the following conditions?

  1. Given any two functions $f_1(x) $ and $f_2(x)$ in the subset, the function $f_1(x)+ f_2(x)$ is also in the subset.
  2. Given any function $f(x)$ in the subset and any real number $c$, the function $cf(x)$ is also in the subset.

You can either guide me to the answer or provide the answer. Whichever would provide the most intuitive solution.

Kevin Perez
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    What have you tried so far? – CyclotomicField Feb 21 '23 at 23:53
  • Well, I do not know how to start with the other sets; however, I have gotten that the monotonic functions satisfy the conditions. – Kevin Perez Feb 22 '23 at 00:01
  • So, show your efforts. (Otherwise the question may be closed.) For example: if $f_1$ and $f_2$ belong to $S_1$, does it follow that $f_1+f_2$ also belongs to $S_1$? Tell us why or why not. – GEdgar Feb 22 '23 at 00:18
  • I do not know how to explicitly prove it, so I need someone to guide me. But Considering functions such as $x^3$,$e^{(-x)}$, all polynomials with degree 1, log functions, trigonometric functions, functions with discontinuities, absolute value functions, and a few other minorities, we see that these functions satisfy the conditions. If I add two linear functions, the result is still linear and monotonic. – Kevin Perez Feb 22 '23 at 00:35
  • @KevinPerez, consider $f_1(x)=x,$ and $f_2(x)=1-x.$ Both are infinitely differentiable. Also, $f_1(x)$ is always increasing, while $f_2(x)$ is always decreasing. Hence, $f_1,f_2\in S_3.$ However, $f_1(x)+f_2(x)=1.$ Hence, $f_1+f_2\notin S_3.$ – aqualubix Feb 22 '23 at 01:07
  • @aqualubix, you gave me an insightful idea, the fastest way I could get this answer would be just to find a counterargument for the statements. And my mistake, I was still right about the monotonicity I just did not consider constants when checking. – Kevin Perez Feb 22 '23 at 01:35
  • @KevinPerez, yes, now you may remove $S_3$ from your considerations. – aqualubix Feb 22 '23 at 01:37
  • @KevinPerez, set $S_1$ satisfies both properties. If $\int_0^3f(x)dx=0,$ we can get that $\int_0^3cf(x)dx=0$ by multiplying by $c$ on both sides. As for the additive property, since the limits are the same, we can just add the definite integrals. – aqualubix Feb 22 '23 at 01:53
  • That is, if $\int_0^3f_1(x)dx=0,$ and $\int_0^3f_2(x)dx=0,$ then $\int_0^3(f_1(x)+f_2(x))dx=0.$ – aqualubix Feb 22 '23 at 01:55
  • @aqualubix I was working on the same one. $S_1$ may satisfy the first condition, If I have an integral with two areas between 0 and 3 that sum to 0 then could I find any other pair that may cancel terms between them? maybe if get a function of the form X + C and X, yet when I was checking desmos no other functions of the form X^d would have a drop below y = 0 innately. – Kevin Perez Feb 22 '23 at 01:57
  • @aqualubix I understand why you said adding the limits on the integrals; however, since we sum the terms inside the functions, what allows you to claim that they will still equal zero when summed? Also, can $S_4$ be manipulated to satisfy the conditions just as you did for $S_1$? – Kevin Perez Feb 22 '23 at 02:02
  • $S_4 $ works for the first condition. if $ (f''(x))^2 = 2f(x)$ then $ \frac{ (f''(x))^2{}{2} = f(x)$ which means when we sum them we will get two equivalent functions that cancel out 2 and leave $(f''(x))^2$ which on the right side would equal $f_1(x) + f_2(x)$ – Kevin Perez Feb 22 '23 at 02:17
  • Let us continue this discussion in chat. – Kevin Perez Feb 22 '23 at 02:18
  • @KevinPerez, check the chat room for my answer to the question you asked me. – aqualubix Feb 22 '23 at 14:23
  • @aqualubix, hey are you good at statistics as well? I have a question that I need some assistance on. – Kevin Perez Mar 12 '23 at 20:13
  • @KevinPerez, well, statistics ain't my thing, but I'll try it out. – aqualubix Mar 13 '23 at 05:49
  • Are you familiar with the incomplete gamma function? – Kevin Perez Mar 13 '23 at 16:13
  • Can you explain why in test 3 of this suite of tests we are using the incomplete gamma function which the chi-squared distribution. – Kevin Perez Mar 13 '23 at 16:15
  • @KevinPerez, sorry man, I don't know any of that stuff. – aqualubix Mar 14 '23 at 04:13
  • @aqualubix how about this one, if f'(x) +f(x) = 1 and f(0) = 2, find f(ln(2)) – Kevin Perez Mar 15 '23 at 23:49
  • @KevinPerez, that is a seperable DE. Letting $y=f(x),$ we get $\frac{dy}{1-y}=dx.$ Integrating both sides, we get that $y=Ce^{-x}+1,$ where $C$ is a constant. But, $y(0)=2.$ This implies that $C=1.$ So, $f(x)=e^{-x}+1.$ This means that $f(\ln(2))=\frac{3}{2}.$ – aqualubix Mar 16 '23 at 05:22
  • That is a good solution, but does that mean the function is a non-autonomous differentiable equation? – Kevin Perez Mar 16 '23 at 15:51
  • And I have another question, I thought differentiable equations have to be in the form g(y)f(x), so does this mean that g(y)f(x) forms a sum when adding like terms that equals to f'(x) + f(x)? Also, your answer is correct. – Kevin Perez Mar 16 '23 at 15:56
  • @KevinPerez, I do not understand what you mean by "differentiable equations have to be of the form $g(y)f(x)"$. Please elaborate. – aqualubix Mar 16 '23 at 16:19
  • @aqualubix, sorry I meant separable DE. – Kevin Perez Mar 16 '23 at 16:21
  • Do you know how to integrate a floor function and a fraction function in the same equation? If you did not know, the integral of the floor function is defined as the sum of integers up to n. – Kevin Perez Mar 16 '23 at 16:57
  • @KevinPerez, yes I've come across integrals like that before, but whether or not I can solve them depends on the question itself. I can try if you send it to me. – aqualubix Mar 17 '23 at 04:52
  • find the value of $\int_{0}^{100}\frac{{x}}{2^{\lfloor x\rfloor}}$ whereby {x} is the fractional part function. – Kevin Perez Mar 17 '23 at 22:35
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    @KevinPerez, I know the answer to that question, but it's too lengthy to type out in the comments. I suggest asking that as a question itself so that you may get insight from others too. The basic idea is to consider what happens to the function in the interval $[n,n+1),$ where $n$ is some whole number. – aqualubix Mar 18 '23 at 09:08
  • does it follow something like this? – Kevin Perez Mar 18 '23 at 19:56
  • I have the solution here, all that is needed at the end is to take the geometric sum formula. – Kevin Perez Mar 18 '23 at 21:16
  • @KevinPerez, yes, your method is identical to mine. – aqualubix Mar 19 '23 at 02:15
  • Ok, well if you are willing. I will give you a free-form set of questions from the same project. If you are doing them you are not allowed to use Taylor series polynomials for the questions and they need to be solved using methods found in calculus 2. This project can be found here. I personally need a little bit of help so it would be good if whenever you had spare time and you wanted to do it, you could. – Kevin Perez Mar 19 '23 at 04:09
  • Also, you might have to use induction for some of them, but it is not meant to be solved in that way. – Kevin Perez Mar 19 '23 at 04:12
  • @KevinPerez, thanks for the problems. I'll try them out. – aqualubix Mar 19 '23 at 08:51
  • @aqualubix, If you have the chance, could you look at this solution and explain to me how y was removed from the integral? here – Kevin Perez Mar 31 '23 at 00:19
  • @KevinPerez, the integrand is not a function of $y.$ Hence, when we evaluate the definite integral, we treat $\frac{e^x}{x}$ as a constant. – aqualubix Mar 31 '23 at 03:29
  • Wow, this was a trivial result. Yet I did not know about this, I believed the dy could have been changed by changing the limits on the integral. For example, if you look at the gaussian distribution, you can change the variables in equation to match other distributions. – Kevin Perez Mar 31 '23 at 03:42
  • @aqualubix A particle is initially at the point(2,0). at t = 0, t starts moving uniformly counterclockwise in a circle of radius 2 such that the particle makes sone complete rotation in 4 seconds. what is the rate of change of the distance between the particle and its initial starting position at t = 2/3? – Kevin Perez Apr 01 '23 at 21:06
  • @aqualubix, also Areeq moves only available to an elliptical island on a lake described by the equation 9x^2 + 25y^2 = 225. Areeq's new home is located at (4,0) and he placed his chicken nugget far, at (-4,0). The water on the island is along the edge of the island. What is the minimum distance Areeq has tot travel each day? – Kevin Perez Apr 01 '23 at 21:11

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