Question

Let $f\left(1\right)=–2$, and $f^{\prime }\left(x\right)\ge 4.2$ for $1\le x\le 6$. The possible value of f(6) lies in the interval

• A. $[19,\infty )$
• B. $[12,15)$
• C. $[15,1a)$
• D. $(-\infty ,12]$

Answer 1

The correct option is A $[19,\infty )$

Given $f\left(1\right)=-2$ and $f^{\prime }\left(x\right)\ge 4.2$ for all $xϵ[1,6]$
$\Rightarrow$ $f^{\prime }\left(x\right)>0\forall xϵ[1,6]$ $\Rightarrow$ f(x) is strictly increasing by l.m.v.t
$\Rightarrow$ $\frac{f\left(6\right)-f\left(1\right)}{6-1}=f^{\prime }\left(x\right)\ge 4.2$
$\Rightarrow$ $f\left(6\right)-f\left(1\right)\ge 21$
$\Rightarrow$ $f\left(6\right)\ge 21+f\left(1\right)$
$\Rightarrow$ $f\left(6\right)\ge 19$
$\Rightarrow$ $f\left(6\right)ϵ[19,\infty )$