If \( f:[0, \pi / 2] \rightarrow[0,1] \) be a differentiable function such that \( f(0)=0 \) and...

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If \( f:[0, \pi / 2] \rightarrow[0,1] \) be a differentiable function such that \( f(0)=0 \) and \( f(\pi / 2)=1 \) then show that:
(a) \( f(\alpha) \cdot f^{\prime}(\alpha)=\frac{1}{\pi} \) for atleast one \( \alpha \in\left(0, \frac{\pi}{2}\right) \)
(b) \( \exists \) atleast one \( \alpha \in\left(0, \frac{\pi}{2}\right) \) such that \( f^{\prime}(\alpha)=\frac{8 a}{\pi^{2}} \)
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