In a reference (O, i, j), the points A (-12, 1150) and B (10, -500). The function f is defined on (-20, 10) by: f(x) = x^ 4 +2.25 x^3-44.5x^2-260x+100


We note Cf his(her,its) representative curve.

We try to determine all the points of intersection of the curve Cf and the line (AB) of coordinates whole.

1.Explain why this problem is to determine all integer(whole) values ​​of k such as vectors AB and AM are collinear, where M is the point of coordinates (k ; k^ 4 +2.25 k^3-44.5k^2-260k+100).

To determine all the points of intersection of the curve Cf and the line (AB) with integer coordinates, we need to find the values of k such that vectors AB and AM are collinear.

Collinearity of two vectors means that they lie on the same line or are parallel to each other. In this case, we want to find points M on the curve Cf whose corresponding vector AM is collinear with the vector AB.

To check for collinearity, we can calculate the slopes of the lines AB and AM. The slope of a line passing through two points (x1, y1) and (x2, y2) can be calculated using the formula:

slope = (y2 - y1) / (x2 - x1)

For the line AB, the points A (-12, 1150) and B (10, -500) can be used. Therefore, the slope of line AB is:

slope_AB = (-500 - 1150) / (10 - (-12)) = -1650 / 22 = -75

Now, we consider the point M on the curve Cf, with coordinates (k, f(k)). The slope of line AM is calculated as:

slope_AM = (f(k) - 1150) / (k - (-12)) = (f(k) - 1150) / (k + 12)

For the vectors AB and AM to be collinear, their slopes must be equal. Therefore, we can set up the equation:

-75 = (f(k) - 1150) / (k + 12)

Solving this equation will give us the integer values of k for which the line AB and curve Cf intersect at points with integer coordinates.