Expert: Ahmed Salami - 10/29/2009

Question
The function f(x)=ax^3 + 3x^2 + bx has a local minimum at x=3 and a point of inflection at x=1 . Find the values of a and b.

Answer
Hi John,
f(x) = ax³ + 3x² + bx
f'(x) = 3ax² + 6x + b
f''(x) = 6ax + 6
If x = 1 is a point of inflection, then f''(1) = 0, i.e
6a(1) + 6 = 0
a = -1
A local minimum occurs when f'(x) = 0, i.e
3ax² + 6x + b = 0
If x = 3 is a local minimum then it satisfies the equation i.e
3a(3)² + 6(3) + b = 0
27a + 18 + b = 0
27(-1) + 18 + b = 0
-9 + b = 0
b = 9

Regards