Question

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. :/

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