Q.8. Let $I(y) = \int_0^1 (xy^2 - y^3) dx$, $y \in C^0[0, 1]$ (see Class #19).
For each $n \ge 1$, let $y_n(x) = \begin{cases} \frac{1}{n} - x & \text{if } x \in [0, \frac{1}{n}] \\ 0 & \text{if } x \in (\frac{1}{n}, 1] \end{cases}$
(i) Show that $y_n \in C^0[0, 1]$, $(\forall n \ge 1)$, and $y_n \to 0$ in $C^0[0, 1]$ as $n \to \infty$.
(ii) Find $I(y_n)$ and use (i) to deduce that $y = 0$ is not a local min of $I$. Hence, $\delta^2 I$ cannot be strongly positive at $y = 0$.
(iii) If $h_n(x) = n y_n(x)$, $(\forall x \in [0, 1], n \ge 1)$, find $||h_n||_0$ and $\delta^2 I(0; h_n)$. Use these calculations to give another proof of the fact that $\delta^2 I$ is not strongly positive at $y = 0$.
