Let's answer questions **12–15** in two steps each, with explanations. --- ## 12. \( g(x) = f(x) + 5,\quad g'() \;\underline{\phantom{>}}\, \) ### Step 1: Find \( g'() \) - \( g'(x) = f'(x) \) - So, \( g'() = f'() \) ### Step 2: Use the Table - \( \) is in interval \( -4 < x < 6 \), where \( f'(x) < \). - Therefore, \( g'() = f'() < \). **Final Answer:** \( g'() < \) --- ## 13. \( g(x) = 3f(x) - 2,\quad g'(-5) \;\underline{\phantom{>}}\, \) ### Step 1: Find \( g'(-5) \) - \( g'(x) = 3f'(x) \) - So, \( g'(-5) = 3f'(-5) \) ### Step 2: Use the Table - \( -5 \) is in interval \( -\infty < x < -4 \), where \( f'(x) > \). - Therefore, \( g'(-5) = 3f'(-5) > \). **Final Answer:** \( g'(-5) > \) --- ## 14. \( g(x) = -f(x),\quad g'(-6) \;\underline{\phantom{>}}\, \) ### Step 1: Find \( g'(-6) \) - \( g'(x) = -f'(x) \) - So, \( g'(-6) = -f'(-6) \) ### Step 2: Use the Table - \( -6 \) is in interval \( -\infty < x < -4 \), where \( f'(x) > \). - Therefore, \( g'(-6) = -f'(-6) < \). **Final Answer:** \( g'(-6) < \) --- ## 15. \( g(x) = f(2x),\quad g'(4) \;\underline{\phantom{>}}\, \) ### Step 1: Find \( g'(4) \) - \( g'(x) = 2f'(2x) \) (by chain rule) - \( g'(4) = 2f'(8) \) ### Step 2: Use the Table - \( 8 \) is in interval \( 6 < x < \infty \), where \( f'(x) > \). - Therefore, \( g'(4) = 2f'(8) > \). **Final Answer:** \( g'(4) > \) --- ### **Summary Table:** | # | Inequality | |----|------------| | 12 | < | | 13 | > | | 14 | < | | 15 | > | Let me know if you want solutions for any of the "Exam Prep" questions!