On a coordinate plane, a curved line labeled f of x with a minimum value of (1.9, negative 5.7) and a maximum value of (0, 2), crosses the x-axis at (negative 0.7, 0), (0.76, 0), and (2.5, 0), and crosses the y-axis at (0, 2).

Question

Which statement is true about the graphed function?

F(x) < 0 over the intervals (-∞, -0.7) and (0.76, 2.5).
F(x) > 0 over the intervals (-∞, -0.7) and (0.76, 2.5).
F(x) < 0 over the intervals (-0.7, 0.76) and (2.5, ∞).
F(x) > 0 over the intervals (-0.7, 0.76) and (0.76, ∞).

Answer 1

To determine which statement is true about the function \( f(x) \), we need to analyze where the function is above or below the x-axis.

Here are the key points given:
- \( f(x) \) has a minimum value at \((1.9, -5.7)\)
- \( f(x) \) has a maximum value at \((0, 2)\)
- \( f(x) \) crosses the x-axis at \( (-0.7, 0) \), \( (0.76, 0) \), and \( (2.5, 0) \)
- \( f(x) \) crosses the y-axis at \( (0, 2) \)

From these points, we can infer the behavior of \( f(x) \) as follows:
1. When \( x < -0.7 \), since the function crosses the x-axis at \( (-0.7, 0) \) and has no maximum or minimum before this point, it must be positive \( f(x) > 0 \).
2. Between \( -0.7 \) and \( 0.76 \) (specifically it goes from \( -0.7 \) to the maximum at \( (0, 2) \) and then down), since it crosses the x-axis at \( (-0.7, 0) \) and \( (0.76, 0) \), the function goes below the x-axis and hence, \( f(x) < 0 \).
3. Between \( 0.76 \) and \( 2.5 \), the function must be below the x-axis again as it has another crossing point at \( (2.5, 0) \), thus \( f(x) < 0 \).
4. For \( x > 2.5 \), the function must be above the x-axis again as the last crossing at \( (2.5, 0) \) had no further minimum or maximum indicated, thus \( f(x) > 0 \).

Therefore, the correct statement is: \
**F(x) < 0 over the intervals (-0.7, 0.76) and (2.5, ∞).**