Let's analyze the question step-by-step.

Problem Overview:

The polynomial function is given as $f(x) = ax^b$, where:

• $a$ is an integer.
• $b$ is a positive integer.

It is stated that: $\lim_{x \to \infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to -\infty} f(x) = -\infty.$

We need to determine which of the statements provided (A, B, C, D) must be true based on the behavior of the function.

Key Observations:

1. Polynomial Degree (Exponent $b$) Behavior:

   • If $b$ is even, $x^b$ will have the same sign for both positive and negative $x$, because: $(-x)^b = x^b \quad \text{(even power)}.$
   • If $b$ is odd, $x^b$ will have opposite signs for positive and negative $x$, because: $(-x)^b = -x^b \quad \text{(odd power)}.$

2. Leading Coefficient $a$:

   • $a > 0$: The function $ax^b$ will have a positive sign for large positive $x$ (if $b$ is even or odd).
   • $a < 0$: The function $ax^b$ will have a negative sign for large positive $x$ (if $b$ is even or odd).

3. Behavior at Limits:

   • The problem states: $\lim_{x \to \infty} f(x) = -\infty, \quad \lim_{x \to -\infty} f(x) = -\infty.$
   • For $f(x) = ax^b$, both limits being negative implies that:
     • $a < 0$, since the negative leading coefficient makes the function trend toward negative infinity for both large positive and large negative $x$.
     • $b$ must be even, because only an even exponent ensures that $x^b$ has the same behavior for both $x \to \infty$ and $x \to -\infty$.

Correct Answer:

The value of $b$ must be even to ensure the same behavior for $x \to \infty$ and $x \to -\infty$, as stated in C.

Therefore, the correct choice is:

C. The value of $b$ must be even, because as $x$ increases without bound or decreases without bound, the end behaviors of $f$ are the same.

Let me know if you have any questions about this solution.

Here are 5 related questions to deepen your understanding:

1. What would change if $\lim_{x \to -\infty} f(x) = \infty$ instead?
2. How does the sign of $a$ affect the graph of a polynomial function?
3. What is the behavior of the function if $b$ is odd but $a > 0$?
4. Why does an even $b$ result in symmetric end behavior for positive and negative $x$?
5. How does the degree of the polynomial affect its dominance as $x \to \pm\infty$?

Tip: Always analyze the parity (odd/even nature) of the exponent when determining end behavior in polynomial functions!