COMEDK Exam 2019 Solved Paper

We have,
‌f(x)={

2a−x	‌	−a<x<a
3x−2a	‌	x≥a

.
‌‌ At ‌x=a
‌LHL=

lim

x→a

(2a−x)=2a−a=a
‌RHL=

lim

x→a

(3x−2a)=3a−2a=a
‌f(a)=3a−2a=a
‌∵LHL=RHL=f(a)‌. ‌
‌ So, f(x) is continuous at x=a.
Now, when x<a,f(x)=2a−x,
which is continuous for all x<a.
Again, when x>a,f(x)=3x−2a, which is continuous for all x>a
So, f(x) is continuous for all x.
Now, at x=a
‌ LHD ‌=

lim

x→a

‌

(2a−x)−a

x−a

‌=

lim

x→a

‌

−(x−a)

(x−a)

=−1
‌ and RHD ‌‌=

lim

x→a

‌

(3x−2a)−a

x−a

‌=

lim

x→a

‌

3(x−a)

x−a

=3
∴‌ LHD ‌‌≠‌ RHD ‌
So, f(x) is not differentiable at x=a.