Question

Let f'(x), be differentiable $\forall x.$ If $f\left(1\right)=-2$ and $f^{\prime }\left(x\right)\ge 2\forall x\epsilon [1,6]$, then

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

Answer 1

Answer: (B)

$f\left(6\right)\ge 8$

f'(x) is differentiable $\forall x\epsilon [1,6]$
By Lagrange's mean value theorem
$f^{\prime }\left(x\right)=\frac{f\left(6\right)-f\left(1\right)}{6-1}$
$f^{\prime }\left(x\right)\ge 2\forall x\epsilon [1,6]$ (given)
$\Rightarrow \frac{f\left(6\right)+2}{5}\ge 2$ $[\because f(1)=-2]$
$\Rightarrow f\left(6\right)\ge 10-2\Rightarrow f\left(6\right)\ge 8$

Hence, option B is correct.