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# Proof Problem 1

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