Why don't derivatives work to find the range of $g (x)=\arcsin x+\arccos x+\arctan x$? In class we were asked to find the range of 

$$g (x)=\arcsin x+\arccos x+\arctan x$$

So to find minimum and maximum, I differentiated the function. $$g^\prime(x)=\frac{1}{1+x^2}$$
Now its maximum value is $1$, so the maximum of $g(x)$ is $\dfrac{3\pi}{4}$, but I couldn't get minima.
Also note that we can't take infinity, as $\arcsin$ and $\arccos$ are involved, which take values from $-1$ to $1$. Why don't we get a minimum? 
Using inverse identities, I know the answer is $\left(\dfrac{\pi}{4}, \dfrac{3\pi}{4}\right)$.

The maximum value of the derivative is $1$, but this doesn't help in finding the maximum and the minimum of the function. What you get from the derivative is that the function is increasing on its domain, which is $[-1,1]$.
So, where and what are the minimum and maximum values?
