Quadratic EquationHard
Question
The largest interval for which x12 - x9 + x4 - x +1 > 0 is
Options
A.- 4 < x ≤ 0
B.0 < x <1
C.-100 < x <100
D.- ∞ < x < ∞
Solution
Given , x12 - x9 + x4 - x +1 > 0
Here, three cass arises
Case I When x ≤ 0
x12 > 0,-x9 > 0, x4 > 0,-x > 0
∴ x12 - x9 + x4 - x + 1 > 0 for all x ≤ 0 .......(i)
Case II When 0 < x ≤ 1
x9 < x4 and x < 1 ⇒ -x9 + x4 > 0 and 1 - x > 0
∴ x12 - x9 + x4 - x + 1 > 0 for all 0 < x ≤ 1 .......(ii)
Case III When x > 1
x12 > x9 and x4 < x
∴ x12 - x9 + x4 - x + 1 > 0 for all x > 1 .......(iii)
∴ From Eqs. (i), (ii) and (iii), the above equation holds for all x ∈ R
Hence, optine (d) is the correct answer.
Here, three cass arises
Case I When x ≤ 0
x12 > 0,-x9 > 0, x4 > 0,-x > 0
∴ x12 - x9 + x4 - x + 1 > 0 for all x ≤ 0 .......(i)
Case II When 0 < x ≤ 1
x9 < x4 and x < 1 ⇒ -x9 + x4 > 0 and 1 - x > 0
∴ x12 - x9 + x4 - x + 1 > 0 for all 0 < x ≤ 1 .......(ii)
Case III When x > 1
x12 > x9 and x4 < x
∴ x12 - x9 + x4 - x + 1 > 0 for all x > 1 .......(iii)
∴ From Eqs. (i), (ii) and (iii), the above equation holds for all x ∈ R
Hence, optine (d) is the correct answer.
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