Question

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

Answer: (B)

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

By Lagrange mean value theorem in [2,4], there exists $c\in$ (2,4)
$f^{\prime }\left(c\right)=\frac{f\left(4\right)-f\left(2\right)}{2}$
Since, $f^{\prime }\left(c\right)\ge 6$
$\Rightarrow f\left(4\right)+4\ge 12$
$\Rightarrow f\left(4\right)\ge 8$