Let $M$ be the space of monotone increasing functions of $[0,1]$ in itself and let $L$ be the space of linear functions over $[0,1]$.
1. For each $f in M$, find the projection of $f$ onto $L$, i.e. the function $f^** in L$ s.t. $int_0^1 (f(x) - f^**(x))^2 text(d) x = min_(phi in L) int_0^1 (f(x) - phi (x) )^2 text(d) x$.
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Hint: Let $phi (x) = a x + b$ and evaluate $a,b$ in such a way that $int_0^1 (f(x) - phi (x))^2 text(d) x$ attains its minimum value.
2. Prove that the functional\( A(f) := \int_0^1 | f(x) - f^*(x)| \text{d} x = \| f - f^* \|_{1,[0,1]}\), which gives the value of the area enclosed by the graphs of $f$ and $f^**$, attains its maximum value over the set $M$.
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Hint: $M$ is a convex cone and $A(f)$ is convex on $M$, therefore it attains its maximum on some extreme point of $M$.
3. Prove that $max_(f in M) A(f) = 1/2$ and try to explicitly find some function $y in M$ s.t. $A(y) = 1/2$.