Question

the function $f(x)$ is defined below. what is the end behavior of $f(x)$?
$f(x) = -1536 + 70x^4 - 16x^5 + 500x^3 - 2x^6 - 2752x - 800x^2$
answer
\bigcirc as $x \to \infty, f(x) \to -\infty$ and
as $x \to -\infty, f(x) \to \infty$
\bigcirc as $x \to \infty, f(x) \to \infty$ and
as $x \to -\infty, f(x) \to \infty$
\bigcirc as $x \to \infty, f(x) \to -\infty$ and
as $x \to -\infty, f(x) \to -\infty$
\bigcirc as $x \to \infty, f(x) \to \infty$ and
as $x \to -\infty, f(x) \to -\infty$

Explanation:

Step1: Find leading term

Leading term is $-2x^6$ (highest degree term).

Step2: Analyze degree and coefficient

Degree=6 (even), coefficient=-2 (negative).

Step3: Determine end behavior

For even degree: $x\to\infty$ or $x\to-\infty$, $x^6\to\infty$. Multiply by -2: $-2x^6\to-\infty$.

Answer:

as $x \to \infty, f(x) \to -\infty$ and as $x \to -\infty, f(x) \to -\infty$