The function f is given by f (x) = –3x6 + 4x2 + x + 8. Which of the following statements is true?
The polynomial f has one global minimum and one global maximum.
The polynomial f has one global minimum.
The polynomial f has no global extrema.
The polynomial f has one global maximum.

To determine the correct statement about the polynomial function \( f(x) = -3x^6 + 4x^2 + x + 8 \), we need to analyze the function's behavior using calculus, specifically by finding its critical points and analyzing the second derivative for concavity.

### Step 1: First Derivative (Critical Points)
The first derivative helps us find the critical points of the function, which are potential locations for local maxima or minima.

\[
f'(x) = \frac{d}{dx}(-3x^6 + 4x^2 + x + 8)
\]
\[
f'(x) = -18x^5 + 8x + 1
\]

We find critical points by setting \( f'(x) = 0 \):

\[
-18x^5 + 8x + 1 = 0
\]

This is a quintic equation (degree 5), and finding exact solutions algebraically can be difficult. However, the general behavior of this equation gives us insight into the number of critical points.

### Step 2: Second Derivative (Concavity)
We now compute the second derivative to analyze concavity and determine whether the critical points correspond to maxima or minima.

\[
f''(x) = \frac{d}{dx}(-18x^5 + 8x + 1)
\]
\[
f''(x) = -90x^4 + 8
\]

### Step 3: Behavior of the Polynomial
- The highest-degree term in \( f(x) \) is \( -3x^6 \), which is a negative even power. This means that as \( x \to \infty \) or \( x \to -\infty \), the function tends toward \( -\infty \).
- Therefore, the function does not have a global maximum because it decreases without bound for large positive and large negative \( x \).

However, since the function is a degree-6 polynomial and exhibits downward opening behavior at the ends, there is likely a **global minimum** somewhere between these regions.

### Conclusion:
The correct statement is:

**The polynomial \( f \) has one global minimum.**

Would you like more details on how the critical points or concavity work here?  

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Here are 5 related questions to explore this concept further:
1. How do you determine local extrema using the first and second derivatives?
2. What is the significance of a polynomial's highest-degree term in its behavior at infinity?
3. How can quintic equations like \( -18x^5 + 8x + 1 = 0 \) be solved numerically?
4. How does the second derivative test help classify critical points?
5. What role does symmetry play in the analysis of polynomial functions?

**Tip:** The sign of the leading coefficient of a polynomial determines whether it opens upwards or downwards.

The function f is given by f (x) = –3x^6 + 4x^2 + x + 8. Which of the following statements is true? The polynomial f has one global minimum and one global maximum. The polynomial f has one global minimum. The polynomial f has no global extrema. The polynomial f has one global maximum.

Explore how to determine global minima and maxima for the polynomial function f(x) = –3x^6 + 4x^2 + x + 8 using calculus techniques. Learn about critical points, concavity, and the second derivative test. Suitable for high school students in Grades 11-12.

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