Expert: Ahmed Salami - 11/12/2008

Question
What values of a and b make f(x) = x^3 + ax^2 + bx have
a)a local maximum at x = -1 and a local minimum at x = 3?
b)a local minimum at x = 4 and a point of inflection at x = 1?

Answer
Hi Davis,
The cubic function f(x) = x^3 + ax^2 + bx has stationary points when f'(x) = 0
f'(x) = 3x^2 + 2ax + b

a) The local maximum and local minimum occurs when
3x^2 + 2ax + b = 0
i.e they are the roots of the quadratic equation
For the quadratic equation px^2 + qx + k = 0,
The sum of the roots = -q/p
and product of roots = k/p
But the roots are -1 and 3, and so we have
-2a/3 = -1 + 3 = 2
b/3 = (-1)(3) = -3
from which we get
a = -3 and b = -9

b)Similarly,
-2a/3 = 4 + 1 = 5
b/3 = (4)(1) = 4
from which we get
a = -15/2 and b = 12

Regards.