There is at least one polynomial function which satisfies the conditions below. F(7) = −28; F'(x) is nonzero; and F"(x) = 0 for all x Give an example of a polynomial function that meets the above criteria.
There is at least one polynomial function which satisfies the conditions below. F(7) = −28; F'(x) is nonzero; and F"(x) = 0 for all x Give an example of a polynomial function that meets the above criteria.
Algebra & Trigonometry with Analytic Geometry
13th Edition
ISBN:9781133382119
Author:Swokowski
Publisher:Swokowski
Chapter4: Polynomial And Rational Functions
Section4.3: Zeros Of Polynomials
Problem 1E
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![### Polynomial Functions and their Characteristics
There is at least one polynomial function which satisfies the conditions below:
\[ F(7) = -28; \, F'(x) \text{ is nonzero}; \text{ and } F''(x) = 0 \text{ for all } x \]
Give an example of a polynomial function that meets the above criteria.
**Explanation:**
- \( F(7) = -28 \): The value of the polynomial function \( F(x) \) at \( x = 7 \) is \(-28\).
- \( F'(x) \text{ is nonzero} \): The first derivative of the polynomial function \( F(x) \) is nonzero.
- \( F''(x) = 0 \text{ for all } x \): The second derivative of the polynomial function \( F(x) \) is zero for all \( x \).
### Example:
An example of a polynomial function that meets these criteria is:
\[ F(x) = 5x - 63 \]
**Verification:**
1. **Check \( F(7) = -28 \):**
\[
F(7) = 5(7) - 63 = 35 - 63 = -28
\]
2. **Check \( F'(x) \text{ is nonzero} \):**
\[
F'(x) = 5
\]
Given that \( F'(x) = 5 \), it is always nonzero for all \( x \).
3. **Check \( F''(x) = 0 \text{ for all } x \):**
\[
F''(x) = 0
\]
Since the second derivative is indeed zero for all \( x \), the function meets all given conditions.
This example demonstrates that \( F(x) = 5x - 63 \) is a suitable polynomial function satisfying the specified conditions.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F323c4818-e3ec-479b-a7ba-1a514193e117%2F2618989e-c02f-4480-8e6b-dfc133a68102%2Fp15upb_processed.png&w=3840&q=75)
Transcribed Image Text:### Polynomial Functions and their Characteristics
There is at least one polynomial function which satisfies the conditions below:
\[ F(7) = -28; \, F'(x) \text{ is nonzero}; \text{ and } F''(x) = 0 \text{ for all } x \]
Give an example of a polynomial function that meets the above criteria.
**Explanation:**
- \( F(7) = -28 \): The value of the polynomial function \( F(x) \) at \( x = 7 \) is \(-28\).
- \( F'(x) \text{ is nonzero} \): The first derivative of the polynomial function \( F(x) \) is nonzero.
- \( F''(x) = 0 \text{ for all } x \): The second derivative of the polynomial function \( F(x) \) is zero for all \( x \).
### Example:
An example of a polynomial function that meets these criteria is:
\[ F(x) = 5x - 63 \]
**Verification:**
1. **Check \( F(7) = -28 \):**
\[
F(7) = 5(7) - 63 = 35 - 63 = -28
\]
2. **Check \( F'(x) \text{ is nonzero} \):**
\[
F'(x) = 5
\]
Given that \( F'(x) = 5 \), it is always nonzero for all \( x \).
3. **Check \( F''(x) = 0 \text{ for all } x \):**
\[
F''(x) = 0
\]
Since the second derivative is indeed zero for all \( x \), the function meets all given conditions.
This example demonstrates that \( F(x) = 5x - 63 \) is a suitable polynomial function satisfying the specified conditions.
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