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.