Question

Let $f$ be the differentiable for all $x$, If $f\left(1\right)=-2$ and $f^{\prime }\left(x\right)\ge 2$ for $[1,6]$, then:

• A. $f\left(6\right)<8$
• B. $f\left(6\right)\ge 8$
• C. $f\left(6\right)=5$
• D. $f\left(6\right)<5$

Answer 1

The correct option is B $f\left(6\right)\ge 8$

Let if possible $f^{\prime }\left(x\right)=2$ for
$\Rightarrow$ $f\left(x\right)=2x+c$ (Integrating both side w.r.t. x)
$\therefore f\left(1\right)=2+c,-2=2+c$
$\Rightarrow c=-4$ $\therefore f\left(x\right)=2x-4$
$\therefore$ $f\left(6\right)=2\times 6-4=8$ $\therefore$ $f\left(6\right)\ge 8$