Let \( f \) is a function, continuous on \( [0,1] \) such that \( f(x) \leq \sqrt{5} \forall x \...

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Let \( f \) is a function, continuous on \( [0,1] \) such that \( f(x) \leq \sqrt{5} \forall x \in[0,1/2] \) and \( f(x) \leq \frac{2}{x} \forall x \in\left[\frac{1}{2}, 1\right] \) then the smallest \( a \) for which \( \int_{0}^{1} f(x) d x \leq a \) holds for all \( f \) is :
(a) \( \sqrt{5} \)
(b) \( \frac{\sqrt{5}}{2}+2 \ln 2 \)
(c) \( 2+\ln \left(\frac{\sqrt{5}}{2}\right) \)
(d) \( 2+2 \ln \left(\frac{\sqrt{5}}{2}\right) \)
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