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 $xϵ[2,4]$, then.

• A. $f\left(4\right)<8$
• B. $f\left(4\right)\ge 8$
• C. $f\left(4\right)>12$
• D. $f\left(4\right)>8$

Answer 1

Answer: (B)

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

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

Applying Lagrange's mean value theorem,

$\frac{f\left(4\right)-f\left(2\right)}{4-2}=f^{\prime }\left(c\right)\ge 6$

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

$\Rightarrow f\left(4\right)-f\left(2\right)\ge 12$

$\Rightarrow f\left(4\right)\ge 12+f\left(2\right)$

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

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