A math question for a test I need quick answer for, its around a function. Could somebody help me?

Question

youre given a function, f:

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

The graph to f has it apex (highest point on the graph) in (1,5).

a) Explain how this gives the two equations:

3a + 2b = 0

a + b = 2

b) Choose a and b.

Answer 1

ax^3 + bx^2 + 3 has vertex (1,5)

take the derivative and set it = 0

3ax^2 +2bx= x(3ax +2b) = 0

x = -2b/3a = 1

3a = -2b

3a +2b = 0

replace x with 1, and set = 5

a(1)^3 + b(1)^2 +3 = 5

a +b =5-3 =2

a+b = 2

3a +2b = 0

2a +2b = 4

a = -4

-4 +b = 2

b = 6

-4x^3 +6x^2 +3 has apex (1,5)