Q!. Decide whether each of the following func are 1-1,onto,homeomorphisms or diffeomorphisms

a. f(x)=X to the power5/3

b. f(x)=X to the power4/3

c. f(x)=3x+5

d. F(x)=e to the power x

e. f(x)=1/x

f. f(x)=1/x to the power 2

Q2. Identify which of the following subsets ofRare closed,open or neither

a. (x,where x is an interger)

b. (x,where x is a rational number)

c. (x,where x=1/n for some natural number n)

d. (x,sin(1/x)=0)

e. (x,xsin(1/x)=0)

f. (x,sin(1/x)>0)

Q3. Sketch the graph of B(x)

Q4. Prove that B'(0)=0

Q5. Inductively prove that B^n(0)=0 for all n.Conclude that B(x) is a c infiniti function.

Q6. modify B(x) to construct a C infiniti function C(x) whcih satisfies

a. C(x) =0 if x is less than or equal to 0

b. C(x) =1 if x is greater that or equal to 1

c. C'(x)>0 if 0<x<1

Q7. Modify C(x) to construct a C infiniti bump function D(x) on the interval [a,b], where D(x) satisfies

a. D(x) =1 for a<x<b

b. D(x) = 0 for x<alpha and x>beta where a<alpha and beta>b

c. D'(x) not equal 0 on the intervals (alpha,a) and (b,beta)

Q8. Use a bump function to construct a diffeomorphism f;[a,b] goes to [c,d] which satisfies f'(a)=f'(b)=1 and f(a) =c,f(b)=d

Please any kind of help would be appreciated.

SK