JEE Advanced | 2014Set, Relation and FunctionHard
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.
Create a free account to view solution
View Solution FreeMore Set, Relation and Function Questions
The set of points where f(x)= is differentiable is...Let ƒ : ℝ→ℝand g :ℝ→ℝ be two non-constant differentiable functions. If ƒ&#...A = {a, e, i, o, u}and B = {i, o} then the true statement is -...If for three disjoint sets A, B, C; n(A) = 10, n(B) = 6 and n(C) = 5, then n(A ∪ B ∪ C) is equal to -...For two data sets, each of size 5, the variances are given to be 4 and 5 and the corresponding means are given to be 2 a...