Consider the function f : 0 , ∞ → R defined by f x = e − log e x . If m and n be respectively the number of points at which f is not continuous and f is not differentiable, then m + n is

Step-by-step solution

We know that, log x is continuous in 0 , ∞ . ⇒ f x = e - log x is continuous in 0 , ∞ Thus, the number of points where f x is discontinuous is 0 . ⇒ m = 0 f x = e log x , 0 < x < 1 e - log x , x ≥ 1 ⇒ f x = x , 0 < x < 1 1 x , x ≥ 1 ⇒ f ' x = 1 , 0 < x < 1 - 1 x 2 , x ≥ 1 ⇒ f ' 1 - = 1 & f ' 1 + = - 1 So, the number of points where f x is non-differentiable is 1 . ⇒ n = 1 ⇒ m + n = 1