1. The problem asks us to find the intervals where the function is increasing based on the given points and line segments. 2. The points are (1,6), (2,3), (4,5), (6,4), (7,6), and (10,8). 3. We analyze the slope between consecutive points to determine if the function is increasing (slope > 0) or decreasing (slope < 0). 4. Between (1,6) and (2,3): slope = \frac{3-6}{2-1} = \frac{-3}{1} = -3 (decreasing). 5. Between (2,3) and (4,5): slope = \frac{5-3}{4-2} = \frac{2}{2} = 1 (increasing). 6. Between (4,5) and (6,4): slope = \frac{4-5}{6-4} = \frac{-1}{2} = -0.5 (decreasing). 7. Between (6,4) and (7,6): slope = \frac{6-4}{7-6} = \frac{2}{1} = 2 (increasing). 8. Between (7,6) and (10,8): slope = \frac{8-6}{10-7} = \frac{2}{3} \approx 0.67 (increasing). 9. Therefore, the function is increasing on intervals where the slope is positive: - From x=2 to x=4 (which includes 2 < x < 5 since 4 < 5) - From x=6 to x=7 - From x=7 to x=10 10. Checking the given intervals: - 5 < x < 7: Between 5 and 6 slope is negative, but between 6 and 7 slope is positive, so partially increasing. - 1 < x < 3: Between 1 and 2 slope is negative, between 2 and 3 slope is positive (since 2 to 4 is increasing), so partially increasing. - 3 < x < 6: Between 3 and 4 slope is positive, between 4 and 6 slope is negative, so partially increasing. - 2 < x < 5: This interval includes 2 to 4 (increasing) and 4 to 5 (part of decreasing segment), so partially increasing. - 6 < x < 8: Between 6 and 7 slope is positive, between 7 and 8 slope is positive, so increasing. - 7 < x < 10: Slope positive, increasing. 11. Final answer: The function is increasing on intervals 2 < x < 5, 6 < x < 8, and 7 < x < 10.