To determine which statement is true about the graphed function \( f(x) \), we analyze the behavior of the function around its x-axis crossings and its maximum and minimum values.
The function crosses the x-axis at the points:
- \( (-0.7, 0) \)
- \( (0.76, 0) \)
- \( (2.5, 0) \)
This means that between these points, the function changes signs.
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Intervals between the x-intercepts and their implications:
- For \( x < -0.7 \): Since there are no x-intercepts in this region and the function heads downward toward \( f(-0.7) \), \( f(x) \) is above the x-axis (positive).
- For \( -0.7 < x < 0.76 \): Here, the function must cross the x-axis from \( (0, 2) \) to \( (0.76, 0) \). Thus, \( f(x) \) is below the x-axis (negative).
- For \( 0.76 < x < 2.5 \): The function goes from crossing the x-axis again and moves toward a maximum, thus remaining below the x-axis until it reaches \( x = 2.5 \) where it returns to zero.
- For \( x > 2.5 \): The function is also going to be above the x-axis (positive) as it diverges away from the maximum.
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Sign analysis:
- Therefore, we conclude:
- \( f(x) > 0 \) on the intervals \( (-\infty, -0.7) \) and \( (2.5, \infty) \).
- \( f(x) < 0 \) on the intervals \( (-0.7, 0.76) \) and \( (0.76, 2.5) \).
- Therefore, we conclude:
Now we compare this analysis with the given statements:
- F(x) < 0 over the intervals (-∞, -0.7) and (0.76, 2.5): False; \( f(x) < 0 \) is only on \( (-0.7, 0.76) \) and part of \( (0.76, 2.5) \).
- F(x) > 0 over the intervals (-∞, -0.7) and (0.76, 2.5): False; \( f(x) < 0 \) between \( (-0.7, 0.76) \).
- F(x) < 0 over the intervals (-0.7, 0.76) and (2.5, ∞): False; \( (2.5, \infty) \) is where \( f(x) > 0 \).
- F(x) > 0 over the intervals (-0.7, 0.76) and (0.76, ∞): False; \( f(x) < 0 \) on \( (-0.7, 0.76) \).
Thus, none of the statements listed accurately reflect the analyzed intervals and the corresponding signs of \( f(x) \).
If you need to revise the options, the correct interpretation would indicate the function is greater than zero outside the intervals where it is negative, and this could lead to a version of specified statement reflecting \( f(x) < 0 \) being true specifically over \( (-0.7, 0.76) \) and part from \( (0.76, 2.5) \), whereas it’s greater than zero for \( x < -0.7 \) and \( x > 2.5 \).