To prove or disprove that $f(n) = 5n^2 + 2n \log n \in \Theta(g(n))$ where $g(n) = n^2$, we need to apply the formal definition of $\Theta$-notation.

Definition of $\Theta(g(n))$:

A function $f(n)$ is said to be in $\Theta(g(n))$ if there are positive constants $c_1, c_2,$ and $n_0$ such that for all $n \geq n_0$: $c_1 \cdot g(n) \leq f(n) \leq c_2 \cdot g(n)$ In other words, $f(n)$ grows asymptotically at the same rate as $g(n)$, bounded both above and below by constant multiples of $g(n)$.

Step 1: Compare the growth rates of $f(n)$ and $g(n)$.

We have:

• $f(n) = 5n^2 + 2n \log n$
• $g(n) = n^2$

We need to check whether $f(n)$ is asymptotically proportional to $g(n) = n^2$. To do this, let's analyze both terms in $f(n)$:

• The term $5n^2$ grows like $n^2$, which is the same as $g(n) = n^2$.
• The term $2n \log n$ grows slower than $n^2$, since $\log n$ grows slower than any polynomial.

Thus, the dominant term in $f(n)$ as $n$ becomes large is $5n^2$, and the term $2n \log n$ becomes negligible compared to $n^2$.

Step 2: Prove the upper bound $f(n) \leq c_2 \cdot g(n)$.

For large $n$, $5n^2$ dominates, so:

$f(n) = 5n^2 + 2n \log n \leq 5n^2 + n^2 \quad (\text{since } n \log n \text{ grows slower than } n^2)$ $f(n) \leq 6n^2$ Thus, we can choose $c_2 = 6$, and there exists an $n_0$ such that for all $n \geq n_0$, $f(n) \leq 6n^2$.

Step 3: Prove the lower bound $f(n) \geq c_1 \cdot g(n)$.

For large $n$, the $5n^2$ term dominates $2n \log n$, so: $f(n) = 5n^2 + 2n \log n \geq 5n^2$ Thus, we can choose $c_1 = 5$, and for sufficiently large $n$, we have $f(n) \geq 5n^2$.

Conclusion:

Since we have shown that: $5n^2 \leq f(n) \leq 6n^2$ for sufficiently large $n$, it follows that $f(n) \in \Theta(n^2)$, meaning: $f(n) \in \Theta(g(n))$

Result:

The claim is proven: $f(n) = 5n^2 + 2n \log n \in \Theta(n^2)$.

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Would you like further clarification on this, or details on specific steps? Here are some related questions:

1. What is the difference between $\Theta$-notation and $O$-notation?
2. How do we analyze the growth of logarithmic terms compared to polynomials?
3. Why is the constant factor ignored in asymptotic analysis?
4. How do we find the precise constants $c_1$ and $c_2$ in asymptotic proofs?
5. How does the growth rate of $n \log n$ compare to other common functions?

Tip: In asymptotic analysis, always focus on the dominant term of a function to determine its growth rate.