Answer is: option3
(C)Solution:
\( f'(x) > 0 \) for \( a < x < c \)
→ The function is increasing on the entire interval from \( a \) to \( c \)
\( f''(x) < 0 \) for \( a < x < b \)
→ The function is concave down from \( a \) to \( b \)
\( f''(x) > 0 \) for \( b < x < c \)
→ The function is concave up from \( b \) to \( c \)
We are looking for a graph that is:
- Increasing from \( a \) to \( c \)
- Concave down from \( a \) to \( b \)
- Concave up from \( b \) to \( c \)
Option A:
- Increasing: Yes
- Concave up from \( a \) to \( b \): No (should be concave down)
- Concave down from \( b \) to \( c \): No (should be concave up)
→ Does not match.
Option B:
- Decreasing from \( a \) to \( c \): No (violates \( f' > 0 \))
→ Does not match.
Option C:
- Increasing: Yes
- Concave down from \( a \) to \( b \): Yes
- Concave up from \( b \) to \( c \): Yes
→ Matches all conditions.
Option D:
- Decreasing from \( a \) to \( c \): No (violates \( f' > 0 \))
→ Does not match.
Final Answer: Option C