Let \(f: [0,1] \to \mathbb R\) be a differentiable function with non-increasing derivative such that \(f(0) = 0 \) and \(f'(1) > 0\). Show that

( A ) \(f(1)\geq f'(1)\), and

( B ) \(\displaystyle \int_{0}^{1}\dfrac{1}{1+f^{2}(x)} \, dx\leq \dfrac{f(1)}{f'(1)}\).

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