Expert: Ahmed Salami - 8/4/2004

Question
Given that: f(x)= (x^3) + (ax^2) +bx has critical points at x=1 and x=3, find a and b and classify the critical points as maxima, minima, or neither.  

Answer
Hi Marissa,
If f(x)= x^3 + ax^2 + bx = y
dy/dx = 3x^2 + 2ax + b
At critical points, dy/dx = 0
Therefore,
3x^2 + 2ax + b = 0  at x = 1 and 3
which gives
3(1)+ 2a(1)+ b = 0
and
3(9)+ 2a(3)+ b = 0
resulting in
2a + b = -3
6a + b = -27
solving simultaneously gives
4a = -24
a = -6
and b = 9

I hope it all helps.
Regards.