DSE 2009 Doubts

[Ayushya Kaul]

Could anyone take a look at question 5 from PART-II of DSE 2009?

5. Consider the function
f(x)= x^2. sin(1/x) if x=/ 0
     =0                  if x=0
Then the following is true about the derivative of f :
a) 1 f )0(' = − and ) f (' x is continuous at x = 0.
b) f )0(' = −1 and ) f (' x is discontinuous at x = 0.
c) f )0(' = 0 and ) f (' x is discontinuous at x = 0.
d) f (' x)is not defined at x = 0.


Basically, I don't know how to solve the LHL and RHL. Could anyone show the steps for it?

[Ayushya Kaul]

I also had a query regarding Q.22 from part II:
http://discussion-forum.2150183.n2.nabble.com/Micro-from-DSE-papers-td7577903.html

[Sinistral]

(5)
f(x) is continuous at x=0. Check with LHL=RHL = value of the function.

f(x) is clearly differentiable. Check with left hand differentiability and right hand differentiability at x=0 to see that f(x) doesn't have a "kink" at x=0.
f'(x)= 2xsin(1/x)-cos(1/x) when x ≠ 0
      = 0                          when x=0

now we need to check the continuity of g(x)=f'(x)
the limit of g(x) is definitely ≠ 0 (infact it doesn't even exist) when x tends to zero. hence g(x) which is nothing but f'(x) is discontinuous at x=0.

[Ayushya Kaul]

Makes sense. Thanks :)

[Ankit]

What is the answer to this question? Is this C?

The key says D but that is definitely wrong. Check http://en.wikipedia.org/wiki/Differentiable_function