Prove that exists $x\in[0,2]$ such that $f(x)=\frac{1}{x}$. [duplicate]

calculus continuity

Let $f$ be a continuous function in the interval $[0,2]$ such that $f(2)=3$. Prove that exists $x\in[0,2]$ such that $f(x)=1/x$.

So I've tried defining a new function $g$ such that $g(x)=f(x)-1/x$ although it didn't lead me anywhere, I tried getting to a situation where I can use the IVT although it seems my attempts haven't gotten me anywhere.

I haven't used any math formatting in my post because I'm not too familiar with the formatting syntax, due to my studies not being in English.

Hopefully if a mod sees my post, they will be able to update my post according to the right formats.

Thanks in advance for any advice!


Solution 1:

Try $g(x) = x f(x)$ and the IVT

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