Consider the following functions : 1. f(x)= array lll 1 x & if & x 0 0 & if & x=0 array . 2. f(x)= array ccc 2 x+5 & if & x 0 x^2+2 x+5 & if & x 0 array . Which of the above functions is/are derivable at x=0 ?

Step-by-step solution

f(x)= cases 1 x & x 0 0 & x=0 cases as there is a discontinuity at x =0 , so function is not differentiable at x =0 aligned & f(x)= array cc 2 x+5 & x 0 x^2+2 x+5 & x 0 array . & f(0)=5 & LHD = _ x 0⁻ f ( x )- f (0) x -0 & = _ x 0 x^2+2 x+5-5 x = _ x 0 x+2=2 & R H D= _ x 0⁺ f(x)-f(0) x-0 & = _ x 0 2 x+5-5 x =2 aligned It is differentiable at x =0 Only (2) is differentiable at x =0