Question

Suppose that $f$ is differentiable for all $x\in R$ and that $f^{\prime }\left(x\right)\le 2$ for all $x$. If $f\left(1\right)=2$ and $f\left(4\right)=8$, then $f\left(2\right)$ has the value equal to

• A. $3$
• B. $4$
• C. $6$
• D. $8$

Answer 1

The correct option is B $4$

Since, $f\left(x\right)$ is differentiable for all $x$. So, differentiable in $(1,2)$.

So, by Lagrange's mean value theorem,

$f^{\prime }\left(c\right)=\frac{f\left(2\right)-f\left(1\right)}{2-1}$

$\Rightarrow f\left(2\right)-2\le 2$

$\Rightarrow f\left(2\right)\le 4$

$f\left(x\right)$ is also differentiable in $(2,4)$.

So, by Lagrange's mean value theorem,

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

$\Rightarrow 8-f\left(2\right)\le 4$

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

Hence, $f\left(2\right)=4$