Sign Interval Fcdd1B

1. **State the problem:** We want to determine the sign of the function $$f(x) = (5x + 1)(4x - 8)(x + 6)$$ on the interval $$-\frac{1}{5} < x < 2$$ given its zeros at $$x = -6$$, $$x = -\frac{1}{5}$$, and $$x = 2$$. 2. **Recall the rule:** The sign of a product depends on the signs of its factors. Between zeros, the sign can only change if a factor changes sign at a zero. 3. **Analyze each factor on the interval $$-\frac{1}{5} < x < 2$$:** - For $$5x + 1$$: zero at $$x = -\frac{1}{5}$$. For $$x > -\frac{1}{5}$$, $$5x + 1 > 0$$. - For $$4x - 8$$: zero at $$x = 2$$. For $$x < 2$$, $$4x - 8 < 0$$. - For $$x + 6$$: zero at $$x = -6$$. For $$x > -6$$, $$x + 6 > 0$$. 4. **Determine the sign of $$f(x)$$ on $$-\frac{1}{5} < x < 2$$:** - $$5x + 1 > 0$$ - $$4x - 8 < 0$$ - $$x + 6 > 0$$ So, $$f(x) = (+) \times (-) \times (+) = -$$. 5. **Conclusion:** On the interval $$-\frac{1}{5} < x < 2$$, $$f(x)$$ is always negative. **Final answer:** B f is always negative on the interval.