Advanced functions

Given the formula: f(x)=x^2+ax+b and the vertex point (3,9)
prove that this is the vertex of a minimum parabola using instantaneous rate of change.

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  1. y = x^2 + ax + b
    dy/dx = 2x + a
    = 0 at the vertex
    so x = -a/2
    but the vertex is (3,9)
    so 3 = -a/2
    a = -6
    also (3,9) satisfies y = x^2 + ax + b, and we know a = -6
    9 = 9 - 6(3) + b
    b = 18

    y = x^2 - 6x + 18
    completing the square:
    y = x^2 - 6x + 9 - 9 + 18
    = (x-3)^2 + 9 , whose vertex is (3,9)

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