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
Let X = {1, 2, 3, 4, 5}. The number of differentordered pairs (Y, Z) that can be formed suchthat and is empty, is :...Let S be a non empty subset of N. Negation of the statement ″There exists a number x∈S such that x is even&#...Let S = {1, 2, 3, 4}. The total number of unordered pairs of disjoint subsets of S is equal to...Let A(h, k), B(1, 1) and C(2, 1) be the vertices of a right angled triangle with AC as its hypotenuse. If the area of th...The median of a set of 9 distinct observations is 20.5. If each of the largest 4 observations of the set is increased by...