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Show that the function \(f : [0, 1] \to \mathbb{R}\), \(f(x) = x\), attains its maximum at \(x = 1\), that this maximum is not an interior point, and that Fermat's theorem does not apply at \(x = 1\) --- compute the available one-sided derivative \(f'_g(1)\) to confirm it need not vanish.
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