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

Answer 1

By integrating f'(x) you obtain f(x)=2x³+2x+C where C is an unknown constant of integration.

In order to evaluate the value of C, you impose the additional constraint (the curve must pass through (-1,2)) and hence f(-1)=2

In this case, f(-1)=2(-1)³+2(-1)+C=-2-2+C=-4+C=2, which implies C=+6.

Finally, f(x) reads f(x)=2x³+2x+6 which satisfies both conditions.

Best,

Davide