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

#Calculus

Easy Math Editor

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Math	Appears as
Remember to wrap math in `\(` ... `\)` or `\[` ... `\]` to ensure proper formatting.
`2 \times 3`	$2 \times 3$
`2^{34}`	$2^{34}$
`a_{i-1}`	$a_{i-1}$
`\frac{2}{3}`	$\frac{2}{3}$
`\sqrt{2}`	$\sqrt{2}$
`\sum_{i=1}^3`	$\sum_{i=1}^3$
`\sin \theta`	$\sin \theta$
`\boxed{123}`	$\boxed{123}$