(5) (10 %) If $f$ is continuous on $[0, 1]$ and if $\int_{0}^{1} f(x) x^n \, dx = 0$ ($n = 0, 1, 2, \ldots$), prove that $f(x) = 0$ on $[0, 1]$.
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(5) (10 %) If $f$ is continuous on $[0, 1]$ and if $\int_{0}^{1} f(x) x^n \, dx = 0$ ($n = 0, 1, 2, \ldots$), prove that $f(x) = 0$ on $[0, 1]$.