Let f : [a, b] → [1, ∞) be a continuous function and let g : ℝ → ℝ be defined as g(x) = cases 0 & if x b. cases Then

JEE Advanced 2014 Mathematics question

Let f : [a, b] → [1, ∞) be a continuous function and let g : ℝ → ℝ be defined as

g(x) =

Then

(A)g(x) is continuous but not differentiable at aCorrect
(B)g(x) is differentiable on ℝ
(C)g(x) is continuous but not differentiable at bCorrect
(D)g(x) is continuous and differentiable at either a or b but not both

Answer: (A),(C)

Solution

g is continuous everywhere: g(a) = 0 matches the left value, and at b both pieces equal . At x = a: left derivative 0, right derivative ; they differ, so g is

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Question ID VK-026288