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MihaiT

#1 h Mar 31, 2025, 6:52 PM

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Let $f,g:R->R$ be continuous, and $f$ is increasing and $g$ is decreasing. If $f,g$ have the same horizontal asymptote at $+\infty$ ,which is the number of solutions of the equation $f(x)=g(x)$ ?(Or can this number not be specified?)

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Etkan

#2 h Mar 31, 2025, 7:27 PM • 1 Y

Y by MihaiT

MihaiT wrote:

Let $f,g:R->R$ be continuous, and $f$ is increasing and $g$ is decreasing. If $f,g$ have the same horizontal asymptote at $+\infty$ ,which is the number of solutions of the equation $f(x)=g(x)$ ?(Or can this number not be specified?)

It depends on whether you consider "increasing" as $x<y\implies f(x)<f(y)$ or as $x<y\implies f(x)\leq f(y)$ .

In the first case, the answer is $0$ . Indeed, having the same horizontal asymptote at $\infty$ means that $\lim \limits _{x\to \infty}f(x)=a=\lim \limits _{x\to \infty}g(x)$ , for some $a\in \mathbb{R}$ . Since $f$ is increasing, we have $f(x)<a$ for all $x\in \mathbb{R}$ , and since $g$ is decreasing, we have $g(x)>a$ for all $x\in \mathbb{R}$ . Hence $f(x)<a<g(x)$ for all $x\in \mathbb{R}$ , and so the number of solutions to $f(x)=g(x)$ is $0$ .

In the second case, the answer is that the number of solutions is either $0$ or infinite (more precisely, either $0$ or $|\mathbb{R}|$ ). The same argument as above gives $f(x)\leq a$ for all $x\in \mathbb{R}$ and $g(x)\geq a$ for all $x\in \mathbb{R}$ . If $f(x)<a$ for all $x\in \mathbb{R}$ or $g(x)>a$ for all $x\in \mathbb{R}$ , there are no solutions to $f(x)=g(x)$ . Else there exist $x_f,x_g\in \mathbb{R}$ such that $f(x_f)=a$ and $g(x_g)=a$ , so $f(x)=a$ for all $x\geq x_f$ (because $f$ is increasing and $f(x)\leq a$ for all $x\in \mathbb{R}$ ) and $g(x)=a$ for all $x\geq x_g$ (because $g$ is decreasing and $g(x)\geq a$ for all $x\in \mathbb{R}$ ). Hence $f(x)=a=g(x)$ for all $x\geq \max \{x_f,x_g\}$ , and so the number of solutions to $f(x)=g(x)$ is $|\mathbb{R}|$ .

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MihaiT

#3 h Apr 1, 2025, 1:27 PM

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Etkan wrote:

MihaiT wrote:

Let $f,g:R->R$ be continuous, and $f$ is increasing and $g$ is decreasing. If $f,g$ have the same horizontal asymptote at $+\infty$ ,which is the number of solutions of the equation $f(x)=g(x)$ ?(Or can this number not be specified?)

It depends on whether you consider "increasing" as $x<y\implies f(x)<f(y)$ or as $x<y\implies f(x)\leq f(y)$ .

In the first case, the answer is $0$ . Indeed, having the same horizontal asymptote at $\infty$ means that $\lim \limits _{x\to \infty}f(x)=a=\lim \limits _{x\to \infty}g(x)$ , for some $a\in \mathbb{R}$ . Since $f$ is increasing, we have $f(x)<a$ for all $x\in \mathbb{R}$ , and since $g$ is decreasing, we have $g(x)>a$ for all $x\in \mathbb{R}$ . Hence $f(x)<a<g(x)$ for all $x\in \mathbb{R}$ , and so the number of solutions to $f(x)=g(x)$ is $0$ .

In the second case, the answer is that the number of solutions is either $0$ or infinite (more precisely, either $0$ or $|\mathbb{R}|$ ). The same argument as above gives $f(x)\leq a$ for all $x\in \mathbb{R}$ and $g(x)\geq a$ for all $x\in \mathbb{R}$ . If $f(x)<a$ for all $x\in \mathbb{R}$ or $g(x)>a$ for all $x\in \mathbb{R}$ , there are no solutions to $f(x)=g(x)$ . Else there exist $x_f,x_g\in \mathbb{R}$ such that $f(x_f)=a$ and $g(x_g)=a$ , so $f(x)=a$ for all $x\geq x_f$ (because $f$ is increasing and $f(x)\leq a$ for all $x\in \mathbb{R}$ ) and $g(x)=a$ for all $x\geq x_g$ (because $g$ is decreasing and $g(x)\geq a$ for all $x\in \mathbb{R}$ ). Hence $f(x)=a=g(x)$ for all $x\geq \max \{x_f,x_g\}$ , and so the number of solutions to $f(x)=g(x)$ is $|\mathbb{R}|$ .

Thanks very much!

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MihaiT

#4 h Apr 2, 2025, 5:30 AM

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MihaiT
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#1VPMMar 31, 2025, 7:43 AM
Given $m_1$ weights, each weighing $k_1$ and another $m_2$ weights with $k_2$ each. Write a algorithm that determines the ways in which a scale can be balanced with a weight $X$ on the left pan, and display the number of possible solutions. (The weights can be placed on both pans and the program starts with the numbers $m_1,k_1,m_2,k_2,X$ . What will be displayed after three successive runs: 5,2,5,1,4 | 5,2,5,1,11 | 5,2,5,1,20?

One answer is possible:
a)10;5;0;
b)20;7;0;
c)20;7;1;
d)10;10;0;
e)10;7;0;
f)20;5;0,

if is possible one hint
Thanks very much!

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