Growth

**mrcheese** · January 17th 2010, 05:04 AM

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)?

Justify your answer briefly."

Many Thanks

**CaptainBlack** · January 17th 2010, 08:51 PM

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)?

Justify your answer briefly."

Many Thanks

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

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

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

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

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

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

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

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

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

CB

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