Consider the following function $f(x)$.
$$f(x) = \int_0^1 \frac{|t - x|}{1 + t^2}\,dt \qquad (0 \leq x \leq 1)$$
(1) Find the real number $\alpha$ satisfying $0 < \alpha < \dfrac{\pi}{4}$ such that $f'(\tan \alpha) = 0$.
(2) For the value of $\alpha$ found in (1), find the value of $\tan \alpha$.
(3) Find the maximum value and minimum value of the function $f(x)$ on the interval $0 \leq x \leq 1$. You may use the fact that $0.69 < \log 2 < 0.7$ if necessary.