= Solution
Let $y\ne0$ be an <eigenfunction>, allowing complex values for the reality argument. The <self-adjoint differential operator> identity and <integration by parts> give
$$
\lambda\int_1^R x^2|y|^2dx=\int_1^R\left[x(x-1)|y'|^2+l(l+1)|y|^2\right]dx.
$$
The boundary term $[x(x-1)\overline y y']_1^R$ vanishes for the same reasons as in part ii. The denominator is real and positive; the numerator is real and strictly positive because $l\geq1$ and $y\ne0$. Hence the <Rayleigh quotient> is
$$
\boxed{\lambda=\frac{\displaystyle\int_1^R[x(x-1)|y'|^2+l(l+1)|y|^2]dx}{\displaystyle\int_1^Rx^2|y|^2dx}\in\mathbb R,\qquad\lambda>0.}
$$
The positivity uses a nondegenerate interval $R>1$.
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