Question

Let $f:R\to R$ be such that $f(0)=0$ and $\left|f^{\prime }(x)\right|\le 5$ 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

Answer: (B)

$\left|f^{\prime }(x)\right|\le 5$
$\underset{0}{\overset{1}{\int }}-5dx\le \underset{0}{\overset{1}{\int }}{f}^{\prime }(x)dx\le \underset{0}{\overset{1}{\int }}5dx$
$\Rightarrow -5\le f(1)-0\le 5$
$\Rightarrow -5\le f(1)\le 5$
$\Rightarrow f(1)\in [-5,5]$