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find the values for a, b, and c such that the function f(x)= x^3 + ax^2 + bx+ c

has a critical point at (1,5) and an inflection point at (2,3).

a= -6
b= 3
c= 7

i got those, but they're wrong. i'm not sure why. :/

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1 answer
  1. f = x^3+ax^2+bx+c
    f' = 3x^2+2ax+b
    f'' = 6x+2a

    for inflection, f''=0, so
    6(2)+2a=0
    a=-6

    for critical point,
    f'=0, so
    3(1)+2(-6)(1)+b=0
    b=9

    f(x) = x^3 - 6x^2 + 9x + c
    5=1-6+9+c
    c=1
    3=8-24+18+c
    c=1

    so,
    f = x^3 - 6x^2 + 9x + 1
    f' = 3x^2-12x+9 so f'(1) = 0
    f'' = 6x-12 so f''(2)=0

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