Question

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
(A) \((5,6)\)
(B) \([-5,5]\)
(C) \((-\infty,-5) \cup(5, \infty)\)
(D) \([-4,4]\)

Answer 1

Ans: (B)
Hint : \(\left|f^{\prime}(x)\right| \leq 5\)
$$
\begin{aligned}
&\int_{0}^{1}-5 d x \leq \int_{0}^{1} f^{\prime}(x) d x \leq \int_{0}^{1} 5 d x \\
&\Rightarrow-5 \leq f(1)-0 \leq 5 \\
&\Rightarrow-5 \leq f(1) \leq 5 \\
&\Rightarrow f(1) \in[-5,5]
\end{aligned}
$$

Answer 2

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