Answer is: option4
15Solution:
We use the Fundamental Theorem of Calculus:
\[ f(b) = f(a) + \int_a^b f'(x)\,dx \]
We go backward from \( x = 6 \) to \( x = 0 \):
\[ f(0) = f(6) - \int_0^6 f'(x)\,dx \]
The integral from 0 to 6 consists of:
- From 0 to 1: area = 3 (below x-axis) → \( -3 \)
- From 1 to 3: area = 7 (above x-axis) → \( +7 \)
- From 3 to 6: area = 10 (below x-axis) → \( -10 \)
\[ \int_0^6 f'(x)\,dx = -3 + 7 - 10 = -6 \]
\[ f(0) = 9 - (-6) = 9 + 6 = \boxed{15} \]
Hence option D