Proving Uniqueness of x in x^2 = xsinx + cosx Using Mean Value Theorem"

[cochemuacos]

Mean value theroem??

Homework Statement

Show that $$x^2 = xsinx + cosx$$ is true only for two values of $$x \in {R}$$

Homework Equations

Intermediate value theorem
Mean value theorem (?)

The Attempt at a Solution

I already know how to prove that there is al least one $$x \in [1,1.5]$$ and another $$x \in [-1.5,-1]$$ where the equation holds. The thing is that I'm not completely sure how to pove that they are unique, I have a geometric argument buy i feel it can be done using the mean value theorem.

Just for you to know, what i did to find out where the x's are, i took $$f(x) = x^2-xsinx-cosx$$ and gave values to the function it turns out that $$f(1) < 0$$ and $$f(1.5) > 0$$ so there must be at leat one $$x \in [1,1.5]$$ where $$f(x) = 0$$ But that's it, I ran out of ideas although i feel I'm really close.

Any ideas or advices will be appreiciated

 

[Mentallic]

Have you noticed that the function is even?