Question

what is the end behavior of the graph of the polynomial function $f(x) = 3x^6 + 30x^5 + 75x^4$?
as $x \to -\infty, y \to -\infty$ and as $x \to \infty, y \to -\infty$.
as $x \to -\infty, y \to -\infty$ and as $x \to \infty, y \to \infty$.
as $x \to -\infty, y \to \infty$ and as $x \to \infty, y \to -\infty$.
as $x \to -\infty, y \to \infty$ and as $x \to \infty, y \to \infty$.

Explanation:

Step1: Identify degree and leading coefficient

The polynomial is $f(x) = 3x^6 + 30x^5 + 75x^4$. Degree $n=6$ (even), leading coefficient $a=3$ (positive).

Step2: Apply end behavior rules

For even degree, $x\to\pm\infty$: $x^6\to\infty$. Multiply by positive $3$: $3x^6\to\infty$. Lower terms become negligible.

Answer:

D. As $x \to -\infty, y \to \infty$ and as $x \to \infty, y \to \infty$.