Suppose that is continuous on  and differentiable on  and that f )0( = ,0 f )1( = ,1 f )2( = .1  (i)

Show that there exists  such that 

Show that there exists  such that 

(iii) Show that there exists  such that .

Check whether the series    is uniformly convergent or not, wher 

Show that the series  does not converge uniformly on the interval 

Give an example for each of the following.

i) A set in  with a unique limit point.

ii) A set in  whose all points except the one are its limit points.

iii) A set having no limit point.

iv) A set S with 

v) A bijection from  to 

Prove that a strictly decreasing function is always one-one

Determine the local minimum and local maximum values of the function f defined by 

If the power series  converges uniformly in  then so does  True or false? Justify.