Set, Relation and FunctionMedium
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
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