FunctionHard
Question
If f(x) = 
x3 + (a - 3) x2 + x + 5 is a one-one function, then
x3 + (a - 3) x2 + x + 5 is a one-one function, thenOptions
A.2 ≤ a ≤ 8
B.1 ≤ a ≤ 2
C.0 ≤ a ≤ 1
D.None of these
Solution
f(x) = 
x3 + (a - 3) x2 + x + 5
f′(x) = (4a - 7) x2 + 2(a - 3) x + 1
D ≤ 0 for all x ∈ R
4(a - 3)2 - 4 (4a - 7) ≤ 0
a2 + 9 - 6a - 4a + 7 ≤ 0
a2 - 10a + 16≤ 0
(a - 8) (a - 2) ≤ 0
or a ∈ 2, 8]
f′(x) is always + ve for a ∈ [2, 8]
x3 + (a - 3) x2 + x + 5f′(x) = (4a - 7) x2 + 2(a - 3) x + 1
D ≤ 0 for all x ∈ R
4(a - 3)2 - 4 (4a - 7) ≤ 0
a2 + 9 - 6a - 4a + 7 ≤ 0
a2 - 10a + 16≤ 0
(a - 8) (a - 2) ≤ 0
or a ∈ 2, 8]
f′(x) is always + ve for a ∈ [2, 8]
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