Find the intervals in which the function $f$ given by $f(x) = x^{2} - 4x + 6$ is:
$(a)$ increasing
$(b)$ decreasing

(A) We have $f(x) = x^{2} - 4x + 6$.
Finding the derivative,we get $f'(x) = 2x - 4$.
Setting $f'(x) = 0$,we have $2x - 4 = 0$,which gives $x = 2$.
The point $x = 2$ divides the real line into two disjoint intervals: $(-\infty, 2)$ and $(2, \infty)$.
$(a)$ In the interval $(2, \infty)$,for any $x > 2$,$f'(x) = 2x - 4 > 0$. Therefore,the function $f$ is strictly increasing in the interval $(2, \infty)$.
$(b)$ In the interval $(-\infty, 2)$,for any $x < 2$,$f'(x) = 2x - 4 < 0$. Therefore,the function $f$ is strictly decreasing in the interval $(-\infty, 2)$.