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Let $f: R \rightarrow R$ be such that $f(0)=0$ and $\left|f^{\prime}(x)\right| \leq 5$ for all $x$. Then $f(1)$ is in
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The correct answer is:
$[-5,5]$
$\left|f^{\prime}(x)\right| \leq 5$
$\int_{0}^{1}-5 d x \leq \int_{0}^{1} f^{\prime}(x) d x \leq \int_{0}^{1} 5 d x$
$\Rightarrow-5 \leq f(1)-0 \leq 5$
$\Rightarrow-5 \leq f(1) \leq 5$
$\Rightarrow f(1) \in[-5,5]$
$\int_{0}^{1}-5 d x \leq \int_{0}^{1} f^{\prime}(x) d x \leq \int_{0}^{1} 5 d x$
$\Rightarrow-5 \leq f(1)-0 \leq 5$
$\Rightarrow-5 \leq f(1) \leq 5$
$\Rightarrow f(1) \in[-5,5]$
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