Which of the following is NOT a possible end behavior for the graph of the function [tex]f(x) = x^6 + 2x^5 + 9x^2[/tex]?

A. [tex]y = x^6[/tex] as [tex]x \to \infty[/tex]
B. [tex]y = x^6[/tex] as [tex]x \to -\infty[/tex]
C. [tex]y = 2x^5[/tex] as [tex]x \to \infty[/tex]
D. [tex]y = 2x^5[/tex] as [tex]x \to -\infty[/tex]

The possible end behavior for the given function, f(x) = x²6 + 2x²5 + 9x²2, is explained. The correct answer, which is Option C, is identified and the reason for its selection is provided.

Explanation:

The possible end behavior for a polynomial function is determined by the highest power of x in the function. In this case, the function f(x) = x6 + 2x5 + 9x2 has a highest power of 6. For a function with an even power, such as 6, the end behavior is the same as the leading term. So, the possible end behaviors are y = x6 as x → ∞ (Option A) and y = x6 as x → -∞ (Option B).

Since the function does not have a term with an exponent of 5, the end behavior y = 2x5 as x → ∞ (Option C) and y = 2x5 as x → -∞ (Option D) are not possible for the given function. Therefore, the correct answer is Option C (y = 2x5 as x → ∞).

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