Consider f ' (x) = 6x^2 + 2 and (-1,2) lies on the graph of f. Find the equation for f(x).

Answer 1

The important concept is that the indefinite integral (antiderivative) gives the function f which satisfies the condition f(-1) = 2

f'(x) =6x² + 2

f(x) = ∫f'(x)dx =2x³ + 2x + c

f(-1) = 2(-1)³ + 2(-1) + c = -2 -2 + c = 2

Thus -4+c =2 and c =6

and f(x) =2x³ + 2x + 6