Definite Integration
General
Grade 12
Question:
If $f(x)$ is a continuous function such that $f(x) \ge 0 \ \forall \ x \in [2,10]$ and $\int_{4}^{8} f(x) dx = 0$, then find $f(6)$.
Step-by-Step Solution
Key Concept: General
$f(x)$ is above the $x$-axis or on the $x$-axis for all $x \in [2,10]$. If $f(x)$ is greater than zero for any sub interval of $[4,8]$, then $\int_{4}^{8} f(x) dx$ must be greater than zero. But $\int_{4}^{8} f(x) dx = 0 \Rightarrow f(x) = 0 \ \forall \ x \in [4,8] \Rightarrow f(6) = 0$.
Correct Answer: 0