Thursday, July 31, 2014

for the cubic: f(x) = ax^3+bx^2+cx+d, a is not zero, find conditions on a,b,c and d to ensure that: a) f is always increasing on (-infinity, +infinity

For f(x) to be increasing, the derivative of f(x) has to be
positive.


We'll determine the first
derivative:


f'(x) = 
(ax^3+bx^2+cx+d)'


f'(x) = 3ax^2 + 2bx +
c


For the expression of the first derivative to be positive, we'll
impose the constraint that the discriminant delta to be
negative.


delta = (2b)^2 - 4*3a*c


delta
= 4b^2 - 12ac


delta < 0


4b^2 -
12ac < 0


We'll divide by 4:


b^2
- 3ac < 0


We'll add 3ac both
sides:


b^2 <
3ac


The constraint for f(x) to be increasing over the
interval (-infinite, +infinite) is that: b^2 < 3ac.

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