= Solution
The left boundary condition is $y(0)=0$, while the right one is $y'(\ell)=-ky(\ell)$. Hence $[yy']_0^\ell=-ky(\ell)^2$. The <Rayleigh quotient> becomes
$$
\lambda\int_0^\ell y^2\,dx=\int_0^\ell\big[(y')^2+qy^2\big]dx+ky(\ell)^2.
$$
Every term on the right is nonnegative. If the whole expression were zero, in particular $\int_0^\ell(y')^2dx=0$, so $y$ would be constant. Its zero value at the left endpoint would force $y\equiv0$, contrary to its being an <eigenfunction>. Therefore
$$
\boxed{\lambda>0.}
$$
This is the positivity mechanism for <Dirichlet-Robin eigenvalues on a finite interval>. The sign of the boundary contribution is positive for the convention $y'+ky=0$ with $k>0$.
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