# Prove that f(x)=c has a solution

## 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:

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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.

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Math_QED
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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##.

• sergey_le and Greg Bernhardt
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.

• Math_QED