Let us analyze the question.

The polynomial function $f(x) = ax^b$, where $a$ is an integer and $b$ is a positive integer, has the following limits: $\lim_{x \to -\infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to \infty} f(x) = -\infty.$

Step 1: Understand the behavior of $f(x)$

The general form $f(x) = ax^b$ depends on the values of $a$ (the coefficient) and $b$ (the power):

1. Sign of $a$: Determines whether the function flips vertically.
   • If $a > 0$, $f(x)$ opens upward.
   • If $a < 0$, $f(x)$ opens downward.
2. Even or Odd $b$: Determines the symmetry and behavior of $f(x)$ at $x \to -\infty$ and $x \to \infty$:
   • If $b$ is even, $x^b$ is positive for all $x$ (same end behavior for $x \to \infty$ and $x \to -\infty$).
   • If $b$ is odd, $x^b$ matches the sign of $x$ (opposite end behavior for $x \to \infty$ and $x \to -\infty$).

Step 2: Use the given limits

From the problem: $\lim_{x \to -\infty} f(x) = -\infty \quad \text{and} \quad \lim_{x \to \infty} f(x) = -\infty.$ This tells us:

• $f(x) \to -\infty$ as $x \to \infty$: This requires $a < 0$, because only negative coefficients lead to $-\infty$ as $x \to \infty$ (since $x^b \to +\infty$ for $b > 0$).
• $f(x) \to -\infty$ as $x \to -\infty$: This happens only if $b$ is even, because for even $b$, $x^b > 0$, and multiplying by $a < 0$ yields $f(x) \to -\infty$.

Step 3: Evaluate the options

• Option A: Incorrect. $a$ must be negative, not positive, to have both end behaviors approach $-\infty$.
• Option B: Correct. $a$ must be negative, and the end behaviors are consistent because $b$ is even.
• Option C: Correct reasoning but incomplete—$b$ being even ensures same end behavior, but $a$ must also be negative, which is not stated.
• Option D: Incorrect. If $b$ were odd, the end behaviors would differ (not both approach $-\infty$).

Final Answer:

B: The value of $a$ must be negative, because as $x$ increases without bound or decreases without bound, the end behaviors of $f$ are different.

Would you like a detailed explanation of why the other options are wrong, or assistance with related polynomial questions? Here are some related questions you might find useful:

1. What happens to $f(x) = ax^b$ if $a > 0$ and $b$ is odd?
2. How do the signs of $a$ and $b$ affect the symmetry of the graph of $f(x)$?
3. What happens to $f(x) = ax^b$ if $b = 0$?
4. Can a polynomial function have opposite end behaviors for $\lim_{x \to \infty}$ and $\lim_{x \to -\infty}$?
5. What role does $b$ (even or odd) play in determining the limits of a polynomial function?

Tip: Always check the end behaviors of the polynomial carefully—they depend on both the coefficient $a$ and the degree $b$.