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a) Let $V \subset \mathbb{R}^{n}$ be a subspace and $Z \in \mathbb{R}^{n}$ a given vector. Find a unit vector $X$ that is perpendicular to $V$ with $\langle X, Z\rangle$ as large as possible.
b) Compute max $\int_{-1}^{1} x^{3} h(x) d x$ where $h(x)$ is any continuous function on the interval $-1 \leq x \leq 1$ subject to the restrictions
$\int_{-1}^{1} h(x) d x=\int_{-1}^{1} x h(x) d x=\int_{-1}^{1} x^{2} h(x) d x=0 ; \quad \int_{-1}^{1}|h(x)|^{2} d x=1 .$
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c) Compute $\min _{a, b, c} \int_{-1}^{1}\left|x^{3}-a-b x-c x^{2}\right|^{2} d x$.
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