Question

find 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) and an inflection point at (2,3).

I got a as -6, but I don't know what b and c are. For b and c I just have b + c= 10. I do not know how to proceed thanks

Answer 1

Inflection point at (2,3) means f"(2)=0
=>
f"(x)=6x+2a = 0 at x=2, or
a=-6(2)/2=-6

Since f(2)=3 (from given point (2,3) ), we conclude that:
f(2)=c+2b-16=3
or
2b+c=19 ....(1)

We also know that there is a critical point at (1,5), so
f'(1)=0, or
b=9
Substituting b=9 in (1) gives
b=9, c=1

Check if f(1)=5.... indeed.

So problem solved.