To determine the sign of F on the interval 0 < x < 3, we can consider the signs of the factors of F(x).
Since F(x) = x^2(x - 3)(x + 1)(x + 5), we can analyze the signs of each factor:
- x^2 is positive for values of x between 0 and 3, since squaring any number gives a positive result.
- (x - 3) is negative for values of x between 0 and 3, since subtracting a positive number decreases the value.
- (x + 1) is positive for values of x between 0 and 3, since adding a positive number increases the value.
- (x + 5) is positive for values of x between 0 and 3, since adding a positive number increases the value.
To determine the overall sign of F on the interval 0 < x < 3, we can multiply the signs of the factors:
Positive x^2 * Negative (x - 3) * Positive (x + 1) * Positive (x + 5) = Negative * Negative * Positive * Positive = Positive
Therefore, F is always positive on the interval 0 < x < 3.
The answer is D. F is always positive on the interval.
F(x) = x^2(x - 3)(x + 1)(x + 5) has zeros at x = -5, x = -1, x = 0, and x = 3. What is the sign of f on the interval 0 < x < 3?
A. F is neither positive nor negative on the interval.
B. F is always negative on the interval.
C. F is sometimes positive and sometimes negative on the interval.
D. F is always positive on the interval.
1 answer