Question

1. the following polynomial function is given by $f(x) = -7x^6 + 2x^2 + 4$. which of the following statements about the end behavior of $f$ is true?

(a) the sign of the leading term of $f$ is positive, and the degree of the leading term of $f$ is even; therefore, $limlimits_{x\to -\infty} f(x) = \infty$ and $limlimits_{x\to \infty} f(x) = \infty$.
(b) the sign of the leading term of $f$ is negative, and the degree of the leading term of $f$ is odd; therefore, $limlimits_{x\to -\infty} f(x) = \infty$ and $limlimits_{x\to \infty} f(x) = -\infty$.
(c) the sign of the leading term of $f$ is positive, and the degree of the leading term of $f$ is odd; therefore, $limlimits_{x\to -\infty} f(x) = -\infty$ and $limlimits_{x\to \infty} f(x) = \infty$.
(d) the sign of the leading term of $f$ is negative, and the degree of the leading term of $f$ is even; therefore, $limlimits_{x\to -\infty} f(x) = -\infty$ and $limlimits_{x\to \infty} f(x) = -\infty$.

Explanation:

Step1: Identify leading term details

The leading term of $f(x) = -7x^6 + 2x^2 + 4$ is $-7x^6$. The leading coefficient is $-7$ (negative), and the degree is $6$ (even).

Step2: Analyze end behavior for even degree

For a polynomial with even degree:

• If $x \to \infty$, $x^6 \to \infty$, so $-7x^6 \to -\infty$, meaning $\lim_{x \to \infty} f(x) = -\infty$.
• If $x \to -\infty$, $(-x)^6 = x^6 \to \infty$, so $-7(-x)^6 = -7x^6 \to -\infty$, meaning $\lim_{x \to -\infty} f(x) = -\infty$.

Step3: Match with options

This matches the description in option (D).

Answer:

(D) The sign of the leading term of $f$ is negative, and the degree of the leading term of $f$ is even; therefore, $\lim_{x \to -\infty} f(x) = -\infty$ and $\lim_{x \to \infty} f(x) = -\infty$.