If the function. $g(x)=\left\{\begin{array}{cc}k \sqrt{x+1}, & 0 \leq x \leq 3 \\ m x+2, & 3<x \leq 5\end{array}\right.$ is differentiable, then the value of $k+m$ is :

JEE Main · 2015 · 04 Apr 2015 · Offline · Continuity and Differentiability

If the function.

g (x) = {k x + 1, m x + 2, 0 \leq x \leq 3 3 < x \leq 5

is differentiable, then the value of

k + m

is :

Answer: (D)

Differentiability demands two things where the pieces join at

x = 3

. Continuity requires

k 4 = 3 m + 2

, i.e.

2 k = 3 m + 2

. Matching slopes requires the left branch's derivative

\frac{k}{2 x + 1} = \frac{k}{4}

x = 3

to equal the right branch's slope

m

, i.e.

k = 4 m

. Substituting,

8 m = 3 m + 2

gives

m = \frac{2}{5}

and

k = \frac{8}{5}

, so

k + m = \frac{8}{5} + \frac{2}{5} = 2

JEE Main · 2015 · 04 Apr 2015 · Offline · Single correct option