Given f ' ' ( x ) = 2 x − 6 and f ' ( − 1 ) = − 2 and f ( − 1 ) = 3 . Find f ' ( x ) = and find f ( 1 ) =

Question

Given 

and  and .

Find    

and find 

Answer 1

Given:

f''(x) = 2x - 6

f'(-1) = -2

f(-1) = 3

Find f'(x) and f(1).

To find f'(x), we have to get the antiderivative of f''(x)

f'(x) = x² - 6x + C

According to given: f'(-1) = -2

f'(-1) = (-1)² - 6(-1) + C = -2

1 + 6 + C = -2

C = -2 - 7

C = -9

Therefore:

f'(x) = x² - 6x - 9

Get the antiderivative of f'(x):

f(x) = (1/3)x³ - 3x² - 9x + C

According to given: f(-1) = 3

Therefore:

f(-1) = (1/3)(-1)³ - 3(-1)² - 9(-1) + C = 3

-1/3 - 3 + 9 + C = 3

17/3 + C = 3

C = 3 - 17/3

C = - 8/3

∴ f(x) = (1/3)x³ - 3x² - 9x - 8/3

f(1) = (1/3)(1)³ - 3(1)² - 9(1) - 8/3

f(1) = 1/3 -3 -9 -8/3

f(1) = -43/3