# problem of increasing function

• October 19th 2010, 09:57 PM
Sambit
problem of increasing function
Find the range of the values for $c$ for which $f(x) = x^3 - 3x^2 + 3cx + 1$ is strictly increasing.
• October 19th 2010, 10:43 PM
Gusbob
Find f'(x). For f(x) to be strictly increasing, f'(x) > 0 for all x.
• October 19th 2010, 10:48 PM
Sambit
yes i know that. but writing $f'(x) > 0$ leads to:-
$3x^2 - 6x + 3c > 0$. after this how can i solve this for $c$ ?
• October 19th 2010, 10:53 PM
Gusbob
Use the quadratic formula, pay attention in particular to the discriminant.

Think about what it means to have f'(x) > 0. It means if you graph f'(x), you should have every single point above the x axis => no real roots.

So under what conditions does a quadratic equation not have real roots?
• October 19th 2010, 10:58 PM
Sambit
a quadratic equation does not have real roots when the discriminant is < zero. so, here, $4 < 4c$, ie, $c>1$. am i correct?
• October 19th 2010, 11:07 PM
Gusbob
yes
• October 19th 2010, 11:12 PM
Sambit
but tell me one thing....here we are considering the non-existence of any real root of the quadratic equation because we have a $> 0$ condition. but in case we had the function decreasing, then also we would have considered the same criterion, that is, the non-existence of any real root; and hence the final answer (ie, c > 1) would be the same.

how can you explain this?
• October 19th 2010, 11:13 PM
Gusbob
If we had a decreasing function (to be more precise, polynomials), you cannot have a positive leading coefficient for for f'(x). Simply because the limit as x gets very large will be positive infinity.
• October 19th 2010, 11:15 PM
Sambit
ok. so the fact is the original equation, ie, $f(x) = x^3 - 3x^2 + 3cx + 1$ can never be MONOTONICALLY DECREASING, but may be decreasing in some interval. is it?

• October 19th 2010, 11:19 PM
Gusbob
Quote:

Originally Posted by Sambit
ok. so the fact is the original equation, ie, $f(x) = x^3 - 3x^2 + 3cx + 1$ can never be MONOTONICALLY DECREASING, but may be decreasing in some interval. is it?

Yes it will decrease at some interval, but never monotonic decreasing.
• October 19th 2010, 11:21 PM
Sambit
ok.. thanks a lot for spending so much time for my thread :D...
• October 19th 2010, 11:23 PM
Gusbob
Don't worry about it. Just another procrastination technique I have.