Science:Math Exam Resources/Courses/MATH105/April 2017/Question 06 (a)/Statement

(a) Determine if there exist two continuous functions $f:[0,1]\to \mathbb {R}$ and $g:[0,1]\to \mathbb {R}$ such that

$\int _{0}^{1}f(x)g(x)dx=0$, but $\int _{0}^{1}f(x)dx\neq 0$ and $\int _{0}^{1}g(x)dx\neq 0$.

If your answer is YES, please give the two functions f and g explicitly by some formulas. If your answer is NO, explain why.