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  4. Let f: R arrow R be such that f(0)=0 and |f'(x)| ≤ 5 for all x. Then f(1) is in

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Q. Let $f: R \rightarrow R$ be such that $f(0)=0$ and $\left|f'(x)\right| \leq 5$ for all $x$. Then $f(1)$ is in

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Solution:

$\left|f'(x)\right| \leq 5$
$\int\limits_{0}^{1}-5 d x \leq \int\limits_{0}^{1} f'(x) d x \leq \int\limits_{0}^{1} 5 d x$
$\Rightarrow -5 \leq f(1)-0 \leq 5$
$\Rightarrow -5 \leq f(1) \leq 5$
$\Rightarrow f(1) \in[-5,5]$