Solution
= Solution
Again \b[$f=0$.] The <change of variables> $t=x^2$ gives
$$
0=\int_0^1 f(x)x^{2n+1}\,dx=\frac12\int_0^1 f(\sqrt t)t^n\,dt\qquad(n\geq0).
$$
Apply part (i) to the <continuous function> $h(t)=f(\sqrt t)$. It vanishes on the whole interval, so does $f$.