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Let f'(x) , be differentiable $\forall x $ . if f(1)=-2 and $f'(x)\geq 2\forall x \in[1,6]$, then 

Answer:

Explanation:

 f'(x)  is differentiable  $\forall x \in[1,6]$

 By Lagrange's mean value theorm

 $f'(x)= \frac{f(6)-f(1)}{6-1}$

 $f'(x)  \geq 2 \forall x \in [1,6]$ (given)

 $\Rightarrow$   $\frac{f(6)+2}{5} \geq 2$       [$\because$  f(1)=-2]

 $\Rightarrow$      $f(6) \geq 10-2 \Rightarrow f(6) \geq 8$