Proving that $\exp(-x^2)$ has a unique fixed point on the interval $[0,1]$ 
Consider the function $g(x)=e^{-x^2}$. Prove that g has a unique fixed point on the interval [0,1]. 

So, our teacher did not go over this section, but assigned it for homework and I have no idea where to even start with a proof. Could someone help me with this please?

Hint: Show that the function is continuous, and is decreasing in the aforementioned interval (by considering the derivative). Then conclude that fixed point has to be unique under these conditions.
Alternatively, look at $f(x)=g(x)-x$. Show that $f(0)$ and $f(1)$ have different signs- and so by intermediate value theorem, $f(x)=0$ for some $x \in [0,1]$, and by considering its derivative, conclude that $f(x)=0$ can hold for only one $x$. 
