Question

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

• A. $f\left(4\right)<8$
• B. $f\left(4\right)\ge 8$
• C. $f\left(4\right)\ge 2$
• D. None of these

Answer 1

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

Since, $f\left(x\right)$ is everywhere continuous and differentiable. Therefore, by Lagrange's meanvalue theorem there exists $C\in (2,4)$ such that
$f^{\prime }\left(c\right)=\frac{f\left(4\right)-f\left(2\right)}{4-2}$
$\Rightarrow \frac{f\left(4\right)+4}{2}\ge 6$ [$f^{\prime }\left(x\right)\ge 6$ for all $x\in [2,4]$]
$\Rightarrow$ $f\left(4\right)\ge 8$