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so f'(x)=0 implies
either both factors must be +ve or both - ve.
if both are negative then
x<1 and x<2 ie common is x<1
if both factors are positive, then,
x-1>0 and x-2>0
i.e. the common is x>2
so we see that function is increasing either if x<1 or if x>2
so if func is increasing then x belongs to open interval minus infinity to 1 union 2 to infinity.
f is decreasing when f'(x)<0
so the product(x-1)(x-2) must be negative
x-1 >0 and (x-2)<0
x-1<0 and x-2>0
continuing like in the first case we get
func is decreasing when 1<x<2