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\displaystyle \int^t_0 (f(x)-f(t))\ dx[br][br]\beginaligned = \displaystyle \int^t_0 f(x)\ dx - t(f(t))[br][br]\beginaligned = t g(t) - t f(t)[br][br]\beginaligned = -t(f(t) - g(t))[br][br]\beginaligned = -t(tg'(t)) = -t^2 g'(t)[br][br]\Rightarrow -t^2 g'(t) >A_0 (t)[br][br]
A_0(t) = x_0 (f(x_0) - f(t))[br][br]\beginaligned = \displaystyle (-1+ \sqrt {1+t})(\frac{1}{ \sqrt {1+t}} - \frac{1}{1+t})[br][br]\beginaligned \Rightarrow A_0(t) = 1- \frac{2}{ \sqrt {1+t}} + \frac{1}{1+t}[br][br]
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