Fine the smallest positive integer n so that
f(x) = x^8 * log(x^3) + x^6 * log(x^5)
is a big-o of x^n

n = 9

Step-by-step explanation:

Let's first prove that for any constants k > 0,

The derivative

and the derivative

Now, applying L'Hôpital's rule we find that

Now, let f be the function

It is easy to see that  f(x) is only if

If  

but both n-8 and n-6 are greater than one, so

and f is

On the other hand, if then  

but 8-n is greater or equal than one, so

and so f(x) in not