Find the values of a,b, and c if the parabola y=a(x^2)+bx+c is tangent to the line y=-2x+3 at (2,-1) and has a critical point when x=3

Answer 1

y' = 2ax + b

we want y'(3) = 0, so
6a+b = 0

the line -2x+3 has slope -2, so
we want y'(2) = -2, so
4a+b = -2

and we have (a,b) = (1,-6)

y = x^2 - 6x + c

We know that y(2) = -1, so

-1 = 4 - 12 + c
so, c = 7

y = x^2 - 6x + 7