Question:
$m$ is the smallest positive integer such that for any integer $n \ge m$, the quantity $n^3 - 7n^2 + 11n - 5$ is positive. What is the value of $m$?
$m$ is the smallest positive integer such that for any integer $n \ge m$, the quantity $n^3 - 7n^2 + 11n - 5$ is positive. What is the value of $m$?
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For cubic inequalities, check boundary integers until the sign becomes permanently positive.
Updated On: Mar 20, 2026
- 4
- 5
- 8
- None of these
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Verified By Collegedunia
The Correct Option is B
Solution and Explanation
Test small $n$:
$n=4$: $64 - 112 + 44 - 5 = -9$ (negative).
$n=5$: $125 - 175 + 55 - 5 = 0$ (non-negative).
$n=6$: $216 - 252 + 66 - 5 = 25$ (positive). For all $n>5$, cubic term dominates, ensuring positivity. Thus $m=5$.
$n=5$: $125 - 175 + 55 - 5 = 0$ (non-negative).
$n=6$: $216 - 252 + 66 - 5 = 25$ (positive). For all $n>5$, cubic term dominates, ensuring positivity. Thus $m=5$.
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