Prove that f(x)=c has a solution

  • Thread starter sergey_le
  • Start date
  • #1
69
14

Homework Statement:

Let function ƒ be f(x)=1/x+sin^2(x) and inff(0,∞)=0.
Prove that every c> 0 has a solution to the equation f(x)=c

Relevant Equations:

-
I'm not sure that inff(0,∞)=0 can help But that was the first section of the question so I thought to point it out anyway.
I'm not sure what I'm supposed to do or what I'm supposed to show.
I was thinking of using the right environment of 0 where f aims for infinity but I don't know how it helps me.
 

Answers and Replies

  • #2
Math_QED
Science Advisor
Homework Helper
2019 Award
1,703
724
Hint: Show that if ##x \to 0+##, then ##f \to +\infty##. Since ##f## is continuous on ##]0,\infty[##, the intermediate value theorem says that ##f(x)=c## must have a solution for ##c > 0##.

You can do something similar to show that ##\inf f((0, + \infty)) = 0##.
 
  • Like
Likes sergey_le and Greg Bernhardt
  • #3
69
14
Hint: Show that if ##x \to 0+##, then ##f \to +\infty##. Since ##f## is continuous on ##]0,\infty[##, the intermediate value theorem says that ##f(x)=c## must have a solution for ##c > 0##.

You can do something similar to show that ##\inf f((0, + \infty)) = 0##.
Thank you.
 
  • Like
Likes Math_QED

Related Threads on Prove that f(x)=c has a solution

Replies
4
Views
2K
  • Last Post
Replies
2
Views
2K
Replies
3
Views
2K
Replies
11
Views
5K
Replies
1
Views
5K
  • Last Post
Replies
1
Views
651
Replies
2
Views
2K
Replies
12
Views
2K
Replies
8
Views
1K
Top