For the function f (x), find the antiderivative whose graph passes through the point M (2; 0;) f (x) = 4x ^ 3 + 2x-4

For the function f (x), we find the antiderivative, the graph of which passes through the point M (2; 0).

f (x) = 4 * x ^ 3 + 2 * x – 4.

1) F (x) = ∫ (4 * x ^ 3 + 2 * x – 4) dx = 4 * x ^ 4/4 + 2 * x ^ 2/2 – 4 * x + C = 4/4 * x ^ 4 + 2/2 * x ^ 2 – 4 * x + C = x ^ 4 + x ^ 2 – 4 * x + C;

2) F (2) = 0;

2 ^ 4 + 2 ^ 2 – 4 * 2 + C = 0;

4 * 4 + 4 – 8 + C = 0;

16 + 4 – 8 + C = 0;

16 – 4 + c = 0;

12 + C = 0;

C = -12;

3) We got the antiderivative of the function passing through the point M (2; 0).

F (x) = x ^ 4 + x ^ 2 – 4 * x + C;

F (x) = x ^ 4 + x ^ 2 – 4 * x – 12.