Consider the following functions:
1. `f(x)={{:((1)/(x),if,xne0),(0,if,x=0):}`
2. `f(x)={{:(2x+5ifxgt0),(x^(2)+2x+5ifxle0):}`
Which of the above functions is / are derivable at x=0?

Options

A. 1 only
B. 2 only
C. Both 1 and 2
D. Neither 1 nor 2

Explanation

`f(x)={{:((1)/(x),xne0),(x,x=0):}`
as there is a discontinuity at x=0, so function is not differentiable at x=0
`f(x)={{:(2x+5,xgt0),(x^(2)+2x+5,xle0):}`
f(0)=5
`LHD=underset(xto0^(-))lim(f(x)-f(0))/(x-0)`
`=underset(xto0)lim(x^(2)+2x+5-5)/(x)=underset(xto0)limx+2=2`
`RHD=underset(xto0^(+))lim(f(x)-f(0))/(x-0)`
`=underset(xto0)lim(2x+5-5)/(x)=2`
`:.` It is differentiable at x=0
`:.` Only (2) is differentiable at x=0