Question

Multiple choice 10 points

Answer 1

Answer:

f'(x) = 5    x<4

      = 2x    4<x<6

     = 4         x>6

Domain is all real numbers except 4 and 6

Step-by-step explanation:

We need to find the derivative of the piecewise function

f'(x) = 5    x<4

      = 2x    4≤x≤6

     = 4         x>6

Is the function continuous at 4

f(4-) = 5(4) - 6 = 20-6 = 14

f(4+) = 4^2 -2 = 16-2 = 14    

this is continuous, so there is no problem with the continuity here

Is the function continuous at

f(6-) = 6^2 -2 = 36-2 = 34

f(6+) = 4(6)+10 = 24+10 = 34

this is continuous, so there is no problem with the continuity here

Now we need to check the derivatives

f'(4-) = 5

f'(4+) = 8

This is not continuous, so we get a piecewise definition for the derivative excluding 4

f'(6-) =12

f'(6+) =4

This is not continuous, so we get a piecewise definition for the derivative excluding 6