1. ## Growth

I have come accross a question which i cant get my head around, but i am guessing it is to do with groth and big o notation?

Could some on point me in the right direction for finding some online material to learn this type of maths? as im lost with it!

Consider the functions

"f1(x) = x3 +3x2 +2x+1 and f2(x) = x4 +12x3 +7x2 +2x+1.
What is the smallest value of m in N such that
f1(x)+ f2(x) ∈ O(xm)?

Many Thanks

2. Originally Posted by mrcheese
I have come accross a question which i cant get my head around, but i am guessing it is to do with groth and big o notation?

Could some on point me in the right direction for finding some online material to learn this type of maths? as im lost with it!

Consider the functions

"f1(x) = x3 +3x2 +2x+1 and f2(x) = x4 +12x3 +7x2 +2x+1.
What is the smallest value of m in N such that
f1(x)+ f2(x) ∈ O(xm)?

Many Thanks
$\displaystyle f_1(x)+f_2(x)$ is a quartic so eventually all the terms other than the one in $\displaystyle x^4$ become negligible compared $\displaystyle x^4$ so:

$\displaystyle f_1(x)+f_2(x)=O(x^4)$

To show that $\displaystyle 4$ is the least positive integer for which:

$\displaystyle f_1(x)+f_2(x)=O(x^m)\$

suppose $\displaystyle m=3$ that tells us that for $\displaystyle x$ sufficiently large there exists a $\displaystyle k>0$:

$\displaystyle |f_1(x)+f_2(x)|<k x^m$

But as $\displaystyle f_1(x)+f_2(x)$ is a quartic:

$\displaystyle \frac{f_1(x)+f_2(x)}{x^4}\le k x^{-1}$

But the left hand side tends to a finite limit as $\displaystyle x \to \infty$ while the right hand side goes to $\displaystyle 0$... etc ... etc

CB