Final answer:
The sign of the function f(x) = (2x - 1)(3x + 5)(x + 1) on the interval -5/3 < x < 1/2 is negative, as determined by evaluating the signs of each factor in the function using a test point within the interval.
Step-by-step explanation:
The student has asked about the sign of the function f(x) = (2x - 1)(3x + 5)(x + 1) on the interval -5/3 < x < 1/2. To find the sign of the function within this interval, we need to evaluate the signs of each factor at values within the interval. The factors change their signs at the roots of the function, which are at x = 1/2, x = -1, and x = -5/3. Since the interval given is from -5/3 < x < 1/2, we will take a test point within this range to determine the sign.
Let's choose x = 0 as a test point because it falls within the interval and makes the calculation straightforward. Evaluating each factor at x = 0 gives us (2(0) - 1) which is negative, (3(0) + 5) which is positive, and ((0) + 1) which is also positive. Multiplying these together: Negative * Positive * Positive = Negative. Therefore, the sign of the function f(x) on the interval -5/3 < x < 1/2 is negative.