CBSE Class 12-science Answered

so f'(x)=0 implies

(x-1)(x-2)>0

so

 either both factors must be +ve or both - ve.

if both are negative then

x-1<0

and

x-2<0

i.e.

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.

next,

f is decreasing when f'(x)<0

so the product(x-1)(x-2) must be negative

so either

x-1 >0 and (x-2)<0

or

x-1<0 and x-2>0

continuing like in the first case we get

func is decreasing when 1<x<2