The function $F(x) = 7x^5$ represents a polynomial function. Here are some key points related to it:

Properties of $F(x) = 7x^5$:

1. Degree: The degree of the function is $5$, as the highest power of $x$ is $5$.
2. Leading Coefficient: The coefficient of the highest degree term ($x^5$) is $7$, making $7$ the leading coefficient.
3. Behavior at Large $|x|$: As $x \to \infty$, $F(x) \to \infty$. As $x \to -\infty$, $F(x) \to -\infty$. This is due to the odd degree and positive leading coefficient.
4. Derivative: The derivative of $F(x) = 7x^5$ is $F'(x) = 35x^4$, which represents the slope of the function at any point $x$.
5. Integral: The indefinite integral of $F(x) = 7x^5$ is: $\int F(x) \, dx = \frac{7x^6}{6} + C,$ where $C$ is the constant of integration.

Would you like further details or examples of applications? 😊

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5 Questions to Expand Understanding:

1. What are the critical points of $F(x) = 7x^5$?
2. How would the graph of $F(x) = 7x^5$ look, and what are its key features?
3. What is the second derivative of $F(x) = 7x^5$, and how does it relate to concavity?
4. Can we evaluate $F(x)$ at specific values like $x = 2$ or $x = -1$?
5. How does $F(x) = 7x^5$ compare to other functions like $G(x) = x^5$ or $H(x) = -7x^5$?

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Tip:

When analyzing polynomial functions, pay close attention to the degree and leading coefficient—they determine the overall shape and end behavior of the graph.