JEE Advanced | 2014Set, Relation and FunctionMedium
Question
Let f : [a, b] → [1, ∞) be a continuous function and let g : R → R be defined as
g(x) =
. Then
g(x) =
. ThenOptions
A.g(x) is continuous but not differentiable at a
B.g(x) is differentiable on R
C.g(x) is continuous but not differentiable at b
D.g(x) is continuous and differentiable at either a or b but not both
Solution
g(a-) = 0, g(a) =
f(t)dt = 0, g(a+) = 
f(t)dt = 0
g(b-) =
f(t)dt = 
f(t)dt
g(b) =f(t)dt = g(b+)
Hence g(x) is continuous at x = a as well as x = b
Now, g′(a-) =
g′(a+) =
= f(a) (≠ 0)
Hence g(x) is not differentiable at x = a.
g′(b-)
= f(b)(≠ 0)
g′(b+)
Hence g(x) is not differentiable at x = b.
f(t)dt = 0, g(a+) = f(t)dt = 0g(b-) =
f(t)dt = f(t)dtg(b) =
f(t)dt = g(b+)Hence g(x) is continuous at x = a as well as x = b
Now, g′(a-) =
g′(a+) =
= f(a) (≠ 0)Hence g(x) is not differentiable at x = a.
g′(b-)
= f(b)(≠ 0)g′(b+)
Hence g(x) is not differentiable at x = b.
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