If f ′(x) = 8 x+12 and f(7) = −4, find f(x)

f'(x) = 8x+12
integrate both sides

you get

f(x) = 4 x^2 +12x + C (C is a constant )

to find C, put x=7 so f(7) = -4

imply -4 = 49 *4 +12 *7 + C
you get C = -284

so f(x) = 4x^2 + 12x -284

antiderivative of f'(x) = 4x^2 + 12x + C

-4 = (4)(49) + (12)(7) + C

C = -284

f(x) = 4x^2 + 12x -284

If f'(x) = 8x + 12 then

f(x) = 4x^2 + 12x + k

f(7) = 4*7^2 + 12*7 + k = 280 + k
280 + k = -4
k = -284

f(x) = 4x^2 + 12x - 284

f(x)=4x^2+12x+C

f(7)=-4=4(7^2)+12(7)+C

-4=4(49)+84+C
-88=196+C

C=-284

so f(x)=4x^2+12x-284