Definite Integration
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Grade 12
Question:
Let $F$ be the set of all continuous real valued functions which are solutions to $f^2(x)=100 + \int_0^x (f(t)f'(t)-f(t)-f'(t)-1) dt$. Find the value of $\frac{1}{|F|} \sum_{f(x)\in F} |f(100)|$.
Step-by-Step Solution
\[ f(0) = \pm 10 \]
Differentiating both sides, we get
\[ \underbrace{(f(x)+1)}_{\text{rejected}}(f'(x)+1)=0 \]
\[ \therefore f(x)=10-x \text{ or } -10-x \]
Hence,
\[ \frac{1}{|F|} \sum_{f(x) \in F} |f(100)| = \frac{1}{2}(90+110)=100 \]
Correct Answer: 100