Question

F(x) = 2(4x + 9)(x - 2)(2x – 9)(x + 5) has zeros at x = -5, x = 0, x = 2, and x = 4. What is the sign of f on the interval 0 < x < 2?

A) f is always positive on the interval.
B) f is always negative on the interval.

Answer 1

The sign of f(x) on the interval 0 < x < 2 is positive.

Step-by-step explanation:

The sign of f(x) on the interval 0 < x < 2 can be determined by analyzing the signs of each factor in f(x). Let's consider each factor separately:

• For the factor (4x + 9), when x is between 0 and 2, 4x is positive and positive, so this factor is positive.
• For the factor (x - 2), when x is between 0 and 2, x is positive and 2 is positive, so this factor is positive.
• For the factor (2x - 9), when x is between 0 and 2, 2x is positive and positive, so this factor is positive.
• For the factor (x + 5), when x is between 0 and 2, x is positive and positive, so this factor is positive.

Since all factors are positive on the interval 0 < x < 2, the sign of f(x) is positive on that interval. Therefore, option A) f is always positive on the interval is the correct answer.