Answer:
f'(x) = 5 x<4
= 2x 4<x<6
= 4 x>6
Domain is all real numbers except 4 and 6
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