strictly increasing

A function is strictly increasing if and only if for all b>a, f(b)>f(a)

Set first derivative=0
and solve for x, find x=4 which is a turning point
Subs x=4 into second derivative
f ' ' (x) = 1>0
so x=4 is a minimum point

Since f(x) is a quadratic equation, all points at the right of the minimum point(including min.) must be in a relation: for all b>a, f(b)>f(a)

It is not monotonic increasing because we can't find any b>a such that f(b)=f(a)

Therefore, f(x) is strictly increasing on [4,infinity)

Yes, f(x) is strictly increasing because
f ' (x) > 0 for all x > 4 and f ' (x) = 0 only if x = 4.