a. Suppose $f(x)=-4 x^{4}$. Which of the following are true? Select all that apply.
As $x \rightarrow \infty, f(x) \rightarrow \infty$
As $x \rightarrow-\infty, f(x) \rightarrow-\infty$
As $x \rightarrow-\infty, f(x) \rightarrow \infty$
As $x \rightarrow \infty, f(x) \rightarrow-\infty$
Answer
\(\boxed{\text{'As } x \rightarrow \infty, f(x) \rightarrow-\infty\text{' and 'As } x \rightarrow-\infty, f(x) \rightarrow-\infty\text{' are true.'}}\)
Step 1 ：Suppose the function is \(f(x)=-4 x^{4}\).
Step 2 ：Since the function \(f(x)=-4 x^{4}\) is a polynomial function of even degree, and the leading coefficient is negative, as \(x\) approaches positive or negative infinity, the function value will approach negative infinity.
Step 3 ：Thus, the statements 'As \(x \rightarrow \infty, f(x) \rightarrow-\infty\)' and 'As \(x \rightarrow-\infty, f(x) \rightarrow-\infty\)' are true.
Step 4 ：\(\boxed{\text{'As } x \rightarrow \infty, f(x) \rightarrow-\infty\text{' and 'As } x \rightarrow-\infty, f(x) \rightarrow-\infty\text{' are true.'}}\)
