If the function `g(X) ={{:( ksqrt ( x+1), 0 le x le 3),( mx+2, 3 lt x le 5):}`is differentiable , then the value of K+ m is

Options

A. `(10)/(3)`
B. 4
C. 2
D. `(16)/(5)`

Answer

C

Explanation

It is given that g(x) is differentiable. So, it is continuous and differentiable at x=3,
Continuity of g(x) is x=3,: If g(x) is continuous at x=3, then
`underset(x to 3^(-))lim g(x)=underset(x to 3^(+))lim (mx+2)=ksqrt(3+1)`
`Rightarrow ksqrt(3+1)=3m+2`
`Rightarrow 2k=3m+2`
Differentiability of g(x) at x=3, If g(x) is continuous at x=3, then
`("LHD of g(x) at x=3")=("RHD of g(x) at x=3")`
`Rightarrow {(d)/(dx) (k+sqrt(x+1))_(x=3)={(d)/(dx)(mx+2)}_(x=3)`
`Rightarrow (k)/(4)=m.....(ii)`
Solving (i) and (ii) we obtain `k=(8)/(5) and m=(2)/(5)`