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 $\left(\dfrac{\pi}{4}, \dfrac{3\pi}{4}\right)$.

You have a strictly increasing function on a bounded interval. Therefore your minimum is at the beginning of the interval and thee maximum at the end.
$$ min = g(-1) = \frac{\pi}{4}$$
$$ max = g(1) = \frac{3 \pi}{4}$$
