Question

Show that${\left(x^2+xy+xlogy\right)}^3\mathbf{\text{is}}O\left(x^6{y}^3\right)$

Short answer

Hence,${\left(x^2+xy+x\log y\right)}^3\text{is}O\left(x^6{y}^3\right)$

Step-by-step solution

Step 1:

Using Big-O notation definition

Let $f(x,y)={\left(x^2+xy+x\log y\right)}^3\text{and}g(x,y)=x^6{y}^3$

Let $k_1=k_2=1$

We know $x>k_1$and$y>k_1$

$\Rightarrow x>1$and y > 1

Step 2:

As x > 1 and y > 1

$\Rightarrow x^2>x$and y > y

Comparing the inequality with $\left|f\right(x,y\left)\right|\ge C\left|g\right(x,y\left)\right|$we get

${\left(x^2+xy+x\log y\right)}^3\text{is}O\left(x^6{y}^3\right)$