x ?5x+6 Is it differentiable?? Is f(x) = x?2 differentiable at x = 2? Justify your answer.

SOLUTION STEP 1 2 (x3)(x2) Given f(x) = x 5x+6= = x 3 x2 x2 STEP 2 We need to find whether f(x)=x-3 is differentiable at x=2. For that ,we consider the left and right hand derivatives of the function at x=2. The left hand derivative of f(x) at x=2 is f(2+h)f(2) f(2) = lim = lim 2+h32+3 = lim h = 1 h2 h h2 h h2 h The right hand derivative of f(x) at x=2 is f(2+h)f(2) 2+h32+3 h f+(2) = lim+ h = lim + h = lim +h = 1 h2 h2 h2 Therefore we have f (2+ = f () Thus the function f(x) is differentiable at x=2. x 5x+6 The graph of f(x) = x2 = x 3 can be drawn as