Let's analyze the function $f(x) = (7x - 2)(5x - 8)(x + 5)(x - 6)$ and the given interval $-5 < x < \frac{2}{7}$.

Step 1: Identify the Zeros of $f(x)$

The zeros are provided as:

• $x = -5$
• $x = \frac{2}{7}$
• $x = \frac{8}{5}$
• $x = 6$

Since we are interested in the interval $-5 < x < \frac{2}{7}$, the important roots here are $x = -5$ and $x = \frac{2}{7}$. These two are the boundary points of the interval, and we need to analyze the sign of $f(x)$ between them.

Step 2: Analyze the Behavior of $f(x)$ in the Interval

The polynomial has four linear factors. We will determine the sign of $f(x)$ by checking the signs of each factor in the interval $-5 < x < \frac{2}{7}$:

• For $(7x - 2)$, at $x = -5$, the factor $7(-5) - 2 = -35 - 2 = -37$ (negative), and at $x = \frac{2}{7}$, the factor is $7(\frac{2}{7}) - 2 = 0$. Thus, this factor is negative in most of the interval but reaches zero at the upper boundary.
• For $(5x - 8)$, at $x = -5$, the factor $5(-5) - 8 = -25 - 8 = -33$ (negative). At $x = \frac{2}{7}$, the factor $5(\frac{2}{7}) - 8 \approx -7.57$ (still negative). This factor remains negative throughout.
• For $(x + 5)$, this becomes $-5 + 5 = 0$ at $x = -5$ and positive for any $x > -5$.
• For $(x - 6)$, this becomes negative at $x < 6$ and remains negative in the interval.

Step 3: Conclusion on the Sign of $f(x)$

From the analysis:

• $(7x - 2)$ is negative in the interval but reaches zero at $x = \frac{2}{7}$.
• $(5x - 8)$ is negative.
• $(x + 5)$ is positive.
• $(x - 6)$ is negative.

Thus, the product of the factors shows that $f(x)$ will change signs within the interval, meaning that $f(x)$ is sometimes positive and sometimes negative on the interval. Therefore, the correct answer is C.

Would you like more details or have any questions?

Here are five related questions to explore further:

1. How does the behavior of a function change when it crosses a zero?
2. What is the importance of determining the sign of a function in intervals?
3. Can you sketch the graph of $f(x)$ based on the zeros and signs of the factors?
4. How do different types of factors (e.g., quadratic vs. linear) affect the sign of the function?
5. How does the multiplicity of zeros impact the behavior of $f(x)$ at those points?

Tip: Always check the signs of individual factors in a polynomial to understand the behavior of the function in various intervals.