Let's analyze each statement carefully:

Statement 1:

True or False: $f(n) = n^3 - 3n^2 + 50n + 11$ is $O(n^3)$

To determine whether $f(n)$ is $O(n^3)$, we need to check if the function $f(n)$ is asymptotically bounded above by $n^3$ as $n$ grows large.

• The function $f(n) = n^3 - 3n^2 + 50n + 11$ has a leading term $n^3$.
• The $O$ notation captures the dominant term as $n$ approaches infinity, ignoring lower-order terms and constants.

Since $n^3$ is the dominant term, the other terms $-3n^2$, $50n$, and $11$ become negligible as $n$ increases. Therefore:

$f(n) = O(n^3)$

Answer: True

Statement 2:

True or False: If $f_1(n)$ is $O(n^3)$ and $\Omega(n^2)$, and $f_2(n)$ is $O(n^4)$ and $\Omega(n)$, then $f(n) = f_1(n) + f_2(n)$ is $O(n^4)$

Let's analyze the given conditions:

• $f_1(n)$ is $O(n^3)$, meaning $f_1(n)$ grows at most like $n^3$ as $n$ increases.
• $f_1(n)$ is $\Omega(n^2)$, meaning $f_1(n)$ grows at least like $n^2$ as $n$ increases.
• $f_2(n)$ is $O(n^4)$, meaning $f_2(n)$ grows at most like $n^4$ as $n$ increases.
• $f_2(n)$ is $\Omega(n)$, meaning $f_2(n)$ grows at least like $n$ as $n$ increases.

Now, consider the sum $f(n) = f_1(n) + f_2(n)$.

• The function $f_2(n)$ is $O(n^4)$, meaning it can grow as large as $n^4$ in the worst case.
• Since the sum of two functions is dominated by the one with the larger asymptotic growth, $f(n)$ will be dominated by $f_2(n)$.

Therefore, $f(n)$ must be $O(n^4)$.

Answer: True

Statement 3:

True or False: $f(n) = n \cdot (\cos n + 1)$ is $\Theta(n)$

Let's analyze the function $f(n) = n \cdot (\cos n + 1)$:

• $\cos(n)$ oscillates between $-1$ and $1$, so $\cos(n) + 1$ oscillates between $0$ and $2$.
• Therefore, $f(n) = n \cdot (\cos(n) + 1)$ oscillates between $0 \cdot n = 0$ and $2 \cdot n = 2n$.

Since $f(n)$ can be expressed as $f(n) = n \cdot (\cos(n) + 1)$ and the multiplicative factor $(\cos(n) + 1)$ is bounded between 0 and 2, the function $f(n)$ grows linearly with $n$, up to a constant factor.

This implies:

$f(n) = \Theta(n)$

Answer: True

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Summary of Answers:

1. True: $f(n) = n^3 - 3n^2 + 50n + 11$ is $O(n^3)$.
2. True: $f(n) = f_1(n) + f_2(n)$ must be $O(n^4)$.
3. True: $f(n) = n \cdot (\cos n + 1)$ is $\Theta(n)$.

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Would you like further details or have any questions?

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Here are some related questions you might find interesting:

1. What does it mean for a function to be $\Theta(n)$ versus $O(n)$?
2. How do lower-order terms affect the time complexity analysis?
3. Can a function be both $O(n^2)$ and $\Omega(n^3)$?
4. What are the implications if $f(n) = \Omega(n^4)$ but not $O(n^4)$?
5. How do oscillating functions like $\cos(n)$ impact time complexity analysis?

Tip: Understanding the difference between $O$, $\Omega$, and $\Theta$ is crucial for analyzing algorithms' performance. $O$ provides an upper bound, $\Omega$ gives a lower bound, and $\Theta$ captures the exact growth rate.