Question

A) Find a second-degree polynomial P such that

P(2) = 7, P'(2) = 9,
and
P''(2) = 6.

B) Find a cubic function
y = ax3 + bx2 + cx + d
whose graph has horizontal tangents at the points
(−2, 9) and (2, 3).

Answer 1

("I" stands for integrate)

P2(x) := ax^0 + bx^1 + cx^2
(P2(x))'' = const

P'' = const = 6

P'(x) = I(P'') = I(6) = 6x + c1
P'(2) = 12 + c1 = 9 --> c1 = -3
P'(x) = 6x - 3

P(x) = I(P') = 3x^2 - 3x + c2
P(2) = 6 + c2 = 7 --> c2 = 1

P(x) = 3x^2 - 3x + 1