Let f(x) be a quadratic expression with positive integral coefficients such that for every α , β ∈ R, β > α , 
. Let g(t) = f ′′ (t) f(t), and g(0) = 12, then –
Text Solution
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(a, c, d)
Let f(x) = ax 2 + bx + c
dx > 0 ⇒ f(x) has no real root
g(0) = f "(0) f(0)
⇒ 12 = 2ac ⇒ ac = 6 also a & c are + ve integers so possible values
a = 6, c = 1; a = 1, c = 6; a = 2, c = 3; a = 3, c = 2
again b 2 < 4ac ⇒ b 2 < 24 so b can be 1, 2, 3, 4
hence possible equations = 4. 4 = 16
also f(x) > 0 ∀ x ∈ R
f(1) = a + b + c minimum f(1) = 2 + 3 + 1 = 6
maximum f(1) = 6 + 4 + 1 = 11
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