Set, Relation and FunctionMedium
Question
If domain & range of function f(x) = 
are same, where a,b ∈ R & b > a, then -
are same, where a,b ∈ R & b > a, then -Options
A.a = - 1 + √2, b = 2 - √2
B.a = 1 + √2, b = 2 - √2
C.a = - 1 - √2, b = 2 + √2
D.a = - 1 + √3, b = 2 - √3
Solution
Domain of f(x) is [a,b]
f′(x) =
= 0 ⇒ x = 
f(a) = f(b) =
& f
= √2
range is
⇒ b - a = a2 & 2(b - a)= b2
∵ a ≠ b
∴ (b2 - a2) = (b - a) ⇒ a + b = 1
⇒ a2 + 2a - 1 = 0
⇒ a = - 1 - √2 & b = 2 + √2 or a = - 1 + √2 & b = 2 - √2
f′(x) =
= 0 ⇒ x = f(a) = f(b) =
& f = √2range is
⇒ b - a = a2 & 2(b - a)= b2
∵ a ≠ b
∴ (b2 - a2) = (b - a) ⇒ a + b = 1
⇒ a2 + 2a - 1 = 0
⇒ a = - 1 - √2 & b = 2 + √2 or a = - 1 + √2 & b = 2 - √2
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