If f(x)= array cc 2 x e^ 1 2 x -3 x e^ -1 2 x e^ 1 2 x +4 e^ -1 2 x & if x 0 0 & if x=0 array . is a real valued function then

Step-by-step solution

Given the function f(x)= array cc 2 x e^ 1 2 x -3 x e^ -1 2 x e^ 1 2 x +4 e^ - 1 2 x & if x 0 0 & if x=0 array . aligned & L.H.D. =f^ (0)= _ h 0 f(0-h)-f(0) -h & = _ h 0 f(-h)-f(0) -h aligned = _ h 0 -2 h e^ -1 2 h +3 h e^ 1 2 h -0 -h (e^ -1 2 h +4 e^ 1 2 h ) .= _ h 0 2 e^ -1 2 h -3 e^ 1 2 h (e^ -1 2 h +4 e^ 1 2 h . )= _ h 0 ( 2 e^ -2 2 h -3 e^ -2 2 h +4 ) aligned & = 0-3 0+4 = -3 4 and, R.H.D. = f ^ (0⁺ ) & = _ h 0 f(0+h)-f(0) h = _ h 0 f(h)-0 h aligned aligned & = 0-3 0+4 = -3 4 and, R.H.D. = f ^ (0⁺ ) & = _ h 0