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Assume that $a \lt b$ and that the continuous function ${\rm f} : \left[a, b\right] \to {\mathbb R}$ has the following two properties: $\quad\left(\rm a\right)~{\rm f}\left(x\right) \geq 0\,, \forall\ x \in \left[a, b\right]\quad$ and $\quad\left(\rm b\right)~\displaystyle{\int_{a}^{b}{\rm f}\left(x\right)\,{\rm d}x = 0}$.

How do I show that ${\rm f}\left(x\right) = 0\,, \forall\ x \in \left[a, b\right]\ {\large\rm }$?

Ian Mateus
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guestfromturkey
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1 Answers1

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$\textbf{Hint:}$ Assume that $f(c) > 0$ for some point $c$. Then continuity implies that $f(x) > 0$ in a small interval around $c$, and with this you should be able to derive a contradiction.

Arthur
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