Solution
= Solution
The trial function $y=x(1-x)$ gives
$$
J[y]=\int_0^1x^3(1-x)^2dx=\frac1{60},\qquad
I[y]=\int_0^1x(1-2x)^2dx+k^2\int_0^1x(1-x)^2dx=\frac16+\frac{k^2}{12}.
$$
Its <Rayleigh quotient> is $5k^2+10$, yielding the upper bound. Conversely, $1/x\geq x$ on $(0,1)$ and the derivative energy is nonnegative, so $I[y]\geq k^2J[y]$ for every admissible $y$. Part (b) therefore gives
$$
\boxed{k\leq z_{k,1}\leq\sqrt{5k^2+10}}.
$$