The Big O estimate for the function
This means that for large values of
grows at the rate of

To find the Big O estimate of the given function
we need to simplify it in terms of its highest order term. Let's break it down step by step.
1. Simplify Each Component:
- First, let's simplify the components of the function:
-

- Similarly,

- The rest of the terms are already in their simplest form.
2. Expand the Product:
- Next, expand the product in the function:
![\[ (3x^2 + 2x\log(x) + 2x^2\log(x))\cdot(\pi x^3 + 2x + 4) \]](https://img.qammunity.org/2024/formulas/social-studies/high-school/lq2qjkmrsuwn9fqhszd7tbe0xrfyoflewf.png)
- This will result in a sum of products, where each term will be a multiplication of terms from each bracket.
3. Identify the Highest Order Term:
- In the expanded form, the highest order term will dominate as x goes to infinity.
- The highest order term will come from the product of the highest order terms in each bracket. In this case, it's

4. Consider the Other Terms:
- The function also has
is the dominant term.
- Comparing
the former is the highest order term because it has both a higher power of x and a logarithmic factor.
5. Big O Estimate:
- The Big O notation focuses on the highest order term for large values of x .
- Therefore, the Big O estimate of f(x) is the highest order term we identified, which is

- However, for Big O notation, we often simplify the coefficient and constants. So the Big O estimate would be
