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 between the curve Cf and the line (AB), we need to find the values of k such that the vectors AB and AM are collinear, where M is the point of coordinates (k, f(k)).
Collinear vectors are parallel or anti-parallel vectors, which means their directions are the same or opposite. If AB and AM are collinear, then the direction of the vector AB will be the same as the direction of the vector AM.
We can find the direction of a vector by calculating its slope. For the vector AB, we can calculate the slope using the formula:
slope_AB = (y2 - y1) / (x2 - x1),
where (x1, y1) and (x2, y2) are the coordinates of points A and B, respectively.
In this case, the coordinates of A are (-12, 1150) and the coordinates of B are (10, -500).
slope_AB = (-500 - 1150) / (10 - (-12))
= (-1650) / 22
= -75
Now, let's calculate the slope of the vector AM. The coordinates of M are (k, f(k)), where f(k) represents the value of the function f at point k.
slope_AM = (f(k) - y1) / (k - x1),
where (x1, y1) are the coordinates of point A.
slope_AM = (f(k) - 1150) / (k - (-12))
= (k^4 + 2.25k^3 - 44.5k^2 - 260k + 100 - 1150) / (k + 12)
To find the collinear points of intersection, we need to find the values of k for which slope_AB = slope_AM:
-75 = (k^4 + 2.25k^3 - 44.5k^2 - 260k + 100 - 1150) / (k + 12)
This equation can be solved to find the values of k for which AB and AM are collinear.