Set, Relation and FunctionHard
Question
If f(x) = ax2[x] - b{x}2, where [.] and {.} denotes greatest integer and fractional part function respectively, then which of the following statements is/are correct -
Options
A.f(x) is continuous at x = 0 ∀ a ∈ R, b = 0
B.f(x) is continuous at x = 0 ∀ b ∈ R, a = 0
C.f′
= 0 for 
= 0 for D.f′′ = 0 for a = b
= 0 for a = bSolution
(A) f(x) = ax2[x]
f(0) = f(0+) = 0 = f(0-)
continuous at x = 0 ∀ a ∈ R
(B) f(x) = - b{x}2
f(0+) = f(0) = 0, f(0-) = - b
continuous when b = 0 only
(C) f(x) = ax2 - b(x-1)2x∈ [1,2)
f′(x) = 2ax - 2b(x - 1)
f′
= 0 ⇒ 3a - b = 0
f″(x) = 2a - 2b
f″ = 0 ⇒ a - b = 0
f(0) = f(0+) = 0 = f(0-)
continuous at x = 0 ∀ a ∈ R
(B) f(x) = - b{x}2
f(0+) = f(0) = 0, f(0-) = - b
continuous when b = 0 only
(C) f(x) = ax2 - b(x-1)2x∈ [1,2)
f′(x) = 2ax - 2b(x - 1)
f′
= 0 ⇒ 3a - b = 0 f″(x) = 2a - 2b
f″
= 0 ⇒ a - b = 0Create a free account to view solution
View Solution FreeMore Set, Relation and Function Questions
The relation R = {(1, 1), (2, 2), (3, 3), (1, 2), (2, 3), (1, 3)} on the set A = {1, 2, 3} is -...Let A be the set of all children in the world and R be a relation in A defined by x R y if x and y have same sex. Then R...A ship is fitted with three engines E1, E2 and E3. The engines function independently of each other with respective prob...The function f : [0, 3] → [1, 29], defined by f(x) = 2x3 - 15x2 + 36x + 1, is...Given function f : R → R is defined by f(x) = .Which of the following holds good ?...