Let $\displaystyle f$be continuous on$\displaystyle [a,b]$. If $\displaystyle f(x)\leq 0$ on $\displaystyle [a,b]$ and $\displaystyle \int_a^b f(x)dx=0$, then show that $\displaystyle f(x)\equiv 0$ on $\displaystyle [a,b]$.
Let $\displaystyle f$be continuous on$\displaystyle [a,b]$. If $\displaystyle f(x)\leq 0$ on $\displaystyle [a,b]$ and $\displaystyle \int_a^b f(x)dx=0$, then show that $\displaystyle f(x)\equiv 0$ on $\displaystyle [a,b]$.
Use continuity! I.E. Suppose $\displaystyle f(x)=c\neq 0$for some $\displaystyle x\in [a,b]$. Then by continuity, there exists a small interval about $\displaystyle x$ in which the value of $\displaystyle f$ on this interval varies by less than, say, $\displaystyle c/2$. What can we say about the integral over this interval?