What are a and b if f(x)=ax^3+bx^2+1 has a point of inflection at (-1,2)?

Since the function is a polynomial, then it is continuous
and differentiable.

Since (-1,2) is an inflection point,
then x = -1 is the root of the 2nd derivative. We'll calculate the first
derivative:

f'(x) = 3ax^2 +
2bx

Now, we'll calculate the second
derivative:

f"(x) = 6ax +
2b

f"(-1) =
0

-6a+2b=0

-6a=-2b

a
= b/3

Since the point of inflection is on the graph, then
(-1,2) verifies the function:

f(-1) =
2

-a+b+1 =
2

-a+b=1

-b/3 +b =
1

-b + 3b = 3

2b =
3

b = 3/2

a =
1/2

The values of a and b are: a = 1/2 and b
= 3/2.