For the function f (x), find the antiderivative, the graph of which has one common point

For the function f (x), find the antiderivative, the graph of which has one common point with the line y, if: f (x) = 4x + 8, y = 3.

First, we find the general form of the antiderivative for this function:
F (x) = 4x ^ 2/2 + 8x + C = 2x ^ 2 + 8x + C is a parabola, branches up. If it has only one common point with the line y = 3, then this point is the vertex of the parabola. Let us first find the abscissa of the vertex of the parabola:
x in = -8 / (2 * 2) = – 8/4 = -2
The vertex lies on the straight line y = 3, therefore the ordinate of the vertex y in = 3, then F (-2) = 3, when substituting we find C:
2 * (- 2) ^ 2 + 8 * (- 2) + C = 3
8-16 + C = 3
C = 3 + 8
C = 11 – we substitute the value of C into the general view of the antiderivative:
F (x) = 2x ^ 2 + 8x + 11 is the desired antiderivative.