Intersection of Functions

Question

There is a cubic equation y=ax^3+bx^2+cx+d and a linear equation y=mx+n, where at a point (-1,14), the linear equation is tangent to the cubic equation and they also intersect at point (2,17). What is the value of a+b+c+d+m+n?

Answer 1

y' = 3ax^2 +2bx + c = (17-14)/(2+1) = 1

linear equation in point slope form is y-17 = (1)(x - 2)

y-17 = x-2

or

y = x +15 where 15 is the y intercept and (2,17) is the intersection point

17 = 8a + 4b + 2c + d after plugging in (2,17)

plug in (-1, 14) too

14 = -a + b -c + d

subtract

3 = 9a +3b +3c

divide by 3

1 = 3a +b+c

m+n = 16

3a +b+c +m +n = 17