Question

Let $f$ be a function which is continuous and differentiable for all real $x$. If $f\left(2\right)=-4$ and $f^{\prime }\left(x\right)\ge 6$ for all $x\in [2,4]$, then

• A. $f\left(4\right)<8$
• B. $f\left(4\right)\ge 8$
• C. $f\left(4\right)\ge 12$
• D. none of these

Answer 1

The correct option is B $f\left(4\right)\ge 8$

By Mean Value Theorem, there exists a real number $c\in (2,4)$ such that

$f^{\prime }\left(c\right)=\frac{f\left(4\right)-f\left(2\right)}{4-2}$

$\Rightarrow f^{\prime }\left(c\right)=\frac{f\left(4\right)+4}{2}$

Since $f^{\prime }\left(x\right)\ge 6\forall x\in [2,4]$

$\therefore f^{\prime }\left(c\right)\ge 6$

$\Rightarrow \frac{f\left(4\right)+4}{2}\ge 6$

$\Rightarrow f\left(4\right)+4\ge 12$

$\Rightarrow f\left(4\right)\ge 8$