Question

Give a big-O estimate for (x² + x(log x)³) · (2x + x3).

Short answer

$\left|f\right(x\left)\right|=x^2+x(logx{)}^3.(2^x+x^3)$is$O\left(x^2{2}^x\right)$

Step-by-step solution

Step 1:

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

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

Where $x>k$

$$\begin{gathered} f\left(x\right)=(x^2+x(logx{)}^3.(2^x+x^3) \\ f\left(x\right)=x^2+x(logx{)}^3+x^5+x^4(logx{)}^3 \\ x>0:(logx{)}^3\le x \\ x>10:x^3\le 2^x \end{gathered}$$

Let us choose $k=10$.For $x>10$we then obtain

$$\begin{gathered} \left|f\right(x\left)\right|=x^2+x(logx{)}^3.(2^x+x^3) \\ =|x^2{2}^x+x{2}^x(logx{)}^3+x^5+x^4(logx{)}^3| \\ =x^2{2}^x+x{2}^x(logx{)}^3+x^5+x^4(logx{)}^3 \\ =\le x^2{2}^x+x{2}^{x}x+x^5+x^{4}x \\ =x^2{2}^x+x{2}^x+x^5+x^5 \\ =x^2{2}^x+x{2}^x+x^2{x}^3+x^2{x}^3 \\ =\le x^2{2}^x+x{2}^x+x^2{2}^x+x^2{2}^x \\ =4x^2{2}^x \\ =4|x^2{2}^x| \end{gathered}$$

C then needs to be choose as at least 4. thus let choose $C=4$

By the definition of the Big-O notation we then obtain that $\left|f\right(x\left)\right|=x^2+x(logx{)}^3.(2^x+x^3)$is $O\left(x^2{2}^x\right)$with $k=10$and $C=4$