Question

Show that 8x³ + 12x + 100 log x is O(x³).

Short answer

$8x^3+12x+100logx$

Step-by-step solution

Step 1:

Big-O notation$f\left(x\right)O\left(g\right(x\left)\right)$ if there exists constsnts C and k such that

$\left|f\right(x\left)\right|\le C\left|g\right(x\left)\right|$

Where$x>k$

To proof:$8x^3+12x+100logx$ is$O\left(x^3\right)$

When$x>k$

Let us choose $k=12$.For$x>12$ we then obtain

$$\begin{gathered} |8x^3+12x+100logx|=8x^3+12x+100logx \\ =8x^3+12.1+10.10.logx \\ =8x^3+x^3+x^3 \\ =10x^3 \\ =10\left|x^3\right| \\ C=10 \end{gathered}$$

C then needs to be chosen as at least 10 thus let us choose$C=10$

By the definition of the Big-O notation we then obtain that$8x^3+12x+100logx$ is$O\left(x^3\right)$ with$k=12$ and$C=10$