# How many solutions to the following equation

#### Mathmatizer

How to prove that tanx=10x has exactly 3 roots in the interval (-pi/2, pi/2)?
I defined g(x) = tanx-10x.
I think I should derive it and use Rolle's theorem and Intermediate value theorem. But I just can't figure out how exactly.

Thanks

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#### romsek

Math Team
Sure looks like infinite roots to me.

Did you mean to restrict the domain?

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#### Mathmatizer

My bad . the domain is restricted to (-pi/2, pi/2) (cannot edit the main post anymore, if a moderator can add the domain to the main post it would be great!)

#### romsek

Math Team
Solve $g'(x)=0$

You'll see there are only 2 solutions on your restricted domain.

Now apply Rolle's theorem.

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