Question

$f\left(x\right)=\{\begin{array}{cc}k\sqrt{x^2+5}& \text{if}-\infty \le x\le 2\\ mx+2& \text{if}2<x\end{array}$
If the given function is differentiable everywhere, then find the value of
$k+m$

• A. $2$
• B. $\frac{8}{5}$
• C. $\frac{2}{5}$
• D. $3$

Answer 1

The correct option is A $2$

$\text{As the function is differentiable}\text{everywhere}$
$⟹\text{It is continuous at all points.}$
$⟹LHS=RHL$
$⟹k\sqrt{4+5}=m\left(2\right)+2$
$⟹3k=2m+2$.............(1)

$\text{As the function is differentiable}$
$⟹LHD=RHD$
$⟹\frac{2k}{3}=m$
$⟹2k=3m$.......................(2)

$\text{Using (1) and (2),}$
$k=\frac{6}{5}\text{and}m=\frac{4}{5}$,
$⟹k+m=2.$