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sachmo

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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

Any kind of insight or Help is appreciated.

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

Any kind of insight or Help is appreciated.

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