Question

This is a Calculus 1 Problem. Show ALL your work giving proper context, and make sureit is written neatly, legibly, and with appropriate organization. Also, MUST SHOW JUSTIFICATION for the problem.

Answer 1

we have that

f(x)=ax^3 --------> interval (-infinite, 2]

f(x)=x^2+b -----> interval (2, infinite)

Remember that

If f is differentiable everywhere

then

the function f must be continuous

so

For x=2

a(2)^3=(2)^2+b

8a=4+b ------> equation 1

Find out the first derivative of each function

f(x)=ax^3 ---------> f'(x)=3ax^2

f(x)=x^2+b --------> f'(x)=2x

For x=2

f'(x)=3ax^2 ------> f'(2)=3a(2)^2=12a

f'(x)=2x ---------> f'(2)=2(2)=4

equate both derivatives

12a=4

a=4/12

a=1/3

substitute in equation 1

8a=4+b

8(1/3)=4+b

b=(8/3)-4

b=-4/3

therefore

a=1/3

b=-4/3