Given The Function F ( X ) = X 4 + 6 X 3 − X 2 − 30 X + 4 F(x)=x^4+6x^3-x^2-30x+4 F ( X ) = X 4 + 6 X 3 − X 2 − 30 X + 4 , Use The Intermediate Value Theorem To Decide Which Of The Following Intervals Contains At Least One Zero. Select All That Apply.A. − 5 , − 4 {-5,-4} − 5 , − 4 B. − 4 , − 3 {-4,-3} − 4 , − 3 C. − 3 , − 2 {-3,-2} − 3 , − 2

Introduction

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that helps us determine the existence of zeros for a given function. In this article, we will use the IVT to decide which of the given intervals contains at least one zero for the polynomial function $f(x)=x^4+6x^3-x^2-30x+4$. We will analyze each interval and determine whether it satisfies the conditions of the IVT.

The Intermediate Value Theorem

The Intermediate Value Theorem states that if a function $f(x)$ is continuous on a closed interval $[a,b]$, and if $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one value $c$ in the interval $(a,b)$ such that $f(c)=k$. In other words, the IVT guarantees that if a function is continuous on a closed interval, it will take on all values between the function values at the endpoints of the interval.

The Given Function

The given function is $f(x)=x^4+6x^3-x^2-30x+4$. We need to determine which of the given intervals contains at least one zero for this function.

Interval A: [-5,-4]

To apply the IVT, we need to evaluate the function at the endpoints of the interval. Let's calculate $f(-5)$ and $f(-4)$.

$f(-5) = (-5)^4 + 6(-5)^3 - (-5)^2 - 30(-5) + 4$ $f(-5) = 625 - 750 - 25 + 150 + 4$ $f(-5) = 4$

$f(-4) = (-4)^4 + 6(-4)^3 - (-4)^2 - 30(-4) + 4$ $f(-4) = 256 - 768 - 16 + 120 + 4$ $f(-4) = -504$

Since $f(-5) = 4$ and $f(-4) = -504$, we have $f(-5) > f(-4)$. The IVT guarantees that there exists at least one value $c$ in the interval $(-5,-4)$ such that $f(c) = 0$. Therefore, interval A contains at least one zero.

Interval B: [-4,-3]

Let's calculate $f(-4)$ and $f(-3)$.

$f(-4) = -504$ (as calculated earlier)

$f(-3) = (-3)^4 + 6(-3)^3 - (-3)^2 - 30(-3) + 4$ $f(-3) = 81 - 486 - 9 + 90 + 4$ $f(-3) = -320$

Since $f(-4) = -504$ and $f(-3) = -320$, we have $f(-4) < f(-3)$. The IVT does not guarantee that there exists a value $c$ in the interval $(-4,-3)$ such that $f(c) = 0$. Therefore, interval B does not contain at least one zero.

Interval C: [-3,-2]

Let's calculate $f(-3)$ and $f(-2)$.

$f(-3) = -320$ (as calculated earlier)

$f(-2) = (-2)^4 + 6(-2)^3 - (-2)^2 - 30(-2) + 4$ $f(-2) = 16 - 96 - 4 + 60 + 4$ $f(-2) = -20$

Since $f(-3) = -320$ and $f(-2) = -20$, we have $f(-3) < f(-2)$. The IVT guarantees that there exists at least one value $c$ in the interval $(-3,-2)$ such that $f(c) = 0$. Therefore, interval C contains at least one zero.

Conclusion

In conclusion, we have used the Intermediate Value Theorem to determine which of the given intervals contains at least one zero for the polynomial function $f(x)=x^4+6x^3-x^2-30x+4$. We found that interval A and interval C contain at least one zero, while interval B does not.

Final Answer

The final answer is:

• Interval A: [-5,-4] contains at least one zero.
• Interval C: [-3,-2] contains at least one zero.
• Interval B: [-4,-3] does not contain at least one zero.
  Intermediate Value Theorem: Frequently Asked Questions =====================================================

Introduction

The Intermediate Value Theorem (IVT) is a fundamental concept in calculus that helps us determine the existence of zeros for a given function. In this article, we will answer some frequently asked questions about the IVT.

Q: What is the Intermediate Value Theorem?

A: The Intermediate Value Theorem states that if a function $f(x)$ is continuous on a closed interval $[a,b]$, and if $k$ is any value between $f(a)$ and $f(b)$, then there exists at least one value $c$ in the interval $(a,b)$ such that $f(c)=k$.

Q: What are the conditions for the Intermediate Value Theorem to hold?

A: The IVT requires that the function $f(x)$ be continuous on a closed interval $[a,b]$. This means that the function must be defined at all points in the interval and that the limit of the function as $x$ approaches any point in the interval must exist.

Q: How do I apply the Intermediate Value Theorem?

A: To apply the IVT, you need to:

1. Evaluate the function at the endpoints of the interval.
2. Determine if the function values at the endpoints are equal or if one is greater than the other.
3. If the function values are not equal, determine if the value you are looking for (e.g. zero) is between the function values at the endpoints.
4. If the value you are looking for is between the function values at the endpoints, then the IVT guarantees that there exists at least one value $c$ in the interval such that $f(c) = k$.

Q: What if the function is not continuous on the interval?

A: If the function is not continuous on the interval, then the IVT does not apply. In this case, you may need to use other methods to determine the existence of zeros for the function.

Q: Can the Intermediate Value Theorem be used to find the maximum or minimum of a function?

A: No, the IVT is only used to determine the existence of zeros for a function. It does not provide information about the maximum or minimum of a function.

Q: Are there any limitations to the Intermediate Value Theorem?

A: Yes, the IVT only guarantees the existence of at least one value $c$ in the interval such that $f(c) = k$. It does not provide information about the number of zeros or the location of the zeros.

Q: Can the Intermediate Value Theorem be used to solve equations?

A: Yes, the IVT can be used to solve equations by finding the zeros of the function. However, it is not a method for solving equations in the classical sense, as it does not provide a specific value for the solution.

Conclusion

In conclusion, the Intermediate Value Theorem is a powerful tool for determining the existence of zeros for a given function. By understanding the conditions for the IVT to hold and how to apply it, you can use this theorem to solve a wide range of problems in calculus and other areas of mathematics.

Final Answer

The final answer is:

• The IVT guarantees the existence of at least one value $c$ in the interval such that $f(c) = k$.
• The IVT requires that the function $f(x)$ be continuous on a closed interval $[a,b]$.
• The IVT can be used to find the zeros of a function, but it does not provide information about the maximum or minimum of a function.
• The IVT can be used to solve equations by finding the zeros of the function.