KCET 2021 Math Solved Paper

f(x)={

x3−1,	1<x<∞
x−1,	−∞<x≤1

We have to check the continuity at x=1.
RHL⇒

lim

x→11+

f(x)=

lim

x→11+

(x3−1)=1−1=0
LHL⇒

lim

x→11+

f(x)=

lim

x→11+

(x−1)=1−1=0
f(1)=1−1=0
Thus the function is continuous at x=1.
f(x)={

3x2,	1<x<∞
1,	−∞<x≤1

Now, check the differentiability at x=1.
LHD at x=1⇒1
RHD at x=1⇒3(1)2=3
As, LHD ≠ RHD, function is not differentiable.