Find the values for a, b, and c so that the function f(x) = x^3 + ax^2 + bx + c has a critical point at (1, 5)

Answer 1

1) 3x^2+2ax+b=0 at x=1 3+2a+b=0
and 5=1+a+b+c
These are the only conditions so you can choose c arbitrarily
2a+b=-3
a+b=4-c
so a= c-7 and b= 11-2c

Answer 2

f(x) = x^3 + ax^2 + bx + c . . . equ 1
f ' (x) = 3x^2 + 2 a x + b . . . equ 2. . . . . differentiate
f ' ' (x) = 6x + 2 a . . . . . . differentiate again
6x + 2 a = 0
a = 3 x . . . . . . . . substitute x = 1
a = 3(1) = 3
f ' (x) = 3x^2 + 2 a x + b = 0 . . . . . from equ 2
3(1)^1 + 2 (3)(1) + b = 0
b = - 9
5 = (1)^3 + (3)(1)^2 - 9 (1) + c . . . . . from equ 1
c= 0
a = 3
b = - 9
f(x) = x^3 + 3x^2 - 9 x