Final answer:
The sign of the polynomial function f(x) = (x - 5)(2x + 7)(7x - 3) on the interval (3/7 ≤ x ≤ 5) is negative.
The answer is option ⇒B. Negative
Step-by-step explanation:
The sign of f(x) on the interval (3/7 ≤ x ≤ 5) can be determined by evaluating the sign of each factor of the polynomial function f(x) = (x - 5)(2x + 7)(7x - 3) at different values of x.
To evaluate the sign, we can choose a test point within each interval and substitute it into the factor. If the result is positive, the factor is positive; if the result is negative, the factor is negative.
Let's evaluate the sign of each factor:
- For (x - 5): If we choose a test point, say x = 4, substituting it into the factor gives (4 - 5) = -1, which is negative.
- For (2x + 7): If we choose a test point, say x = 4, substituting it into the factor gives (2(4) + 7) = 15, which is positive.
- For (7x - 3): If we choose a test point, say x = 4, substituting it into the factor gives (7(4) - 3) = 25, which is positive.
Since one factor is negative and the other two factors are positive, the overall sign of f(x) on the interval (3/7 ≤ x ≤ 5) is Negative (B).
The answer is option ⇒B. Negative