The graph of a derivative fX is shown in the figure above Fi

The graph of a derivative f\'(X) is shown in the figure above. Fill in the table of values for f(x) given that f (0) = 9. Enter the exact answers.

Solution

First finding the function f from [0,1]

y - 0 = -6(x-0)

y = -6x, which is the equation of f\'(x)

f(x) = -3x^2 + C

Using the initial condition

f(0) = -3(0)^2 + C

C = 9

Hence f(1) = -3(1)^2 + 9 = 6

For the function [1,3]

f\'(x) = -6

f(x) = -6x + C

f(1) = -6(1) + P

P = 12

f(x) = 12 - 6x

f(2) = 12 - 6(2) = 0

f(3) = 12 - 6(3) = -6

For the function [3,5]

f\'(x) = 6x

f(x) = 3x^2 + P

f(3) = 3(3)^2 + P

P = -33

f(4) = 3(4)^2 - 33 = 15

f(5) = 3(5)^2 - 33 = 42

For the function [3,5]

f\'(x) = 6

f(x) = 6x + Q

f(5) = 6(5) + Q

Q = 12

f(x) = 6x + 12

f(6) = 6(6) + 12 = 48