Question

Suppose that $f$ is differentiable for all $x$ 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

Answer: (B)

$4$

Since, $f$ is differentiable for all $x$.
$\Rightarrow f$ is differentiable in $(1,4)$
Also, $f$ is continuous on $[1,4]$
So, by Lagrange mean value theorem in $[1,4]$, there exists $c\in (0,4)$ such that
$f^{\prime }\left(c\right)=\frac{f\left(4\right)-f\left(1\right)}{3}$
$f^{\prime }\left(c\right)=2$
$\Rightarrow f^{\prime }\left(x\right)=2$
$\Rightarrow f\left(x\right)=2x+k$
$f\left(1\right)=2+k$
$k=0$
$\Rightarrow f\left(x\right)=2x$
$\Rightarrow f\left(2\right)=4$