To solve these problems, we need to determine between which consecutive integers the real zeros of the given functions are located, based on the Intermediate Value Theorem. This theorem tells us that if a function is continuous on an interval $[a, b]$ and changes sign (from positive to negative or vice versa) between two points, there must be at least one real zero between those points.

Let’s go through each function and interval step by step:

Problem 13: $f(x) = x^3 - x^2 - 3$, interval $[-2, 4]$

We will evaluate $f(x)$ at each integer point within the interval $[-2, 4]$ and observe where the sign changes.

Problem 14: $g(x) = -x^3 + 6x + 2$, interval $[-4, 4]$

Similarly, evaluate $g(x)$ at each integer point in the interval $[-4, 4]$ and determine where the sign changes.

Problem 15: $f(x) = 2x^4 - 3x^3 + x^2 - 3$, interval $[-3, 3]$

For this quartic function, we will evaluate $f(x)$ at each integer in $[-3, 3]$ and find where the function changes signs, indicating the presence of zeros.

I'll calculate the values of the functions at each integer point within the given intervals for these three problems to identify where the real zeros are located. Let me begin calculating those values.Here are the values of the functions at each integer point within the given intervals, along with where sign changes occur (indicating zeros between consecutive integers):

Problem 13: $f(x) = x^3 - x^2 - 3$, interval $[-2, 4]$

• $f(-2) = -15$
• $f(-1) = -5$
• $f(0) = -3$
• $f(1) = -3$
• $f(2) = 1$ ← Sign change (zero between 1 and 2)
• $f(3) = 15$
• $f(4) = 45$

Zero located between $x = 1$ and $x = 2$.

Problem 14: $g(x) = -x^3 + 6x + 2$, interval $[-4, 4]$

• $g(-4) = 42$
• $g(-3) = 11$
• $g(-2) = -2$ ← Sign change (zero between -3 and -2)
• $g(-1) = -3$
• $g(0) = 2$ ← Sign change (zero between -1 and 0)
• $g(1) = 7$
• $g(2) = 6$
• $g(3) = -7$ ← Sign change (zero between 2 and 3)
• $g(4) = -38$

Zeros located between:

• $x = -3$ and $x = -2$
• $x = -1$ and $x = 0$
• $x = 2$ and $x = 3$.

Problem 15: $f(x) = 2x^4 - 3x^3 + x^2 - 3$, interval $[-3, 3]$

• $f(-3) = 249$
• $f(-2) = 57$
• $f(-1) = 3$
• $f(0) = -3$ ← Sign change (zero between -1 and 0)
• $f(1) = -3$
• $f(2) = 9$ ← Sign change (zero between 1 and 2)
• $f(3) = 87$

Zeros located between:

• $x = -1$ and $x = 0$
• $x = 1$ and $x = 2$.

Would you like more details or explanations on any part of the solution?

Here are 5 questions related to this problem:

1. What is the Intermediate Value Theorem, and how does it help find zeros?
2. How can we refine the search for zeros using numerical methods like Newton's Method?
3. Why do cubic and quartic functions sometimes have multiple real roots?
4. What is the significance of a sign change in determining real zeros?
5. How can derivatives help identify potential zeros or extrema?

Tip: Always verify if a sign change occurs at integer points. If there’s no sign change, there is no zero in that interval.