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 $m$?

• A. 4
• B. 5
• C. 8
• D. None of these

Hint:

For cubic inequalities, check boundary integers until the sign becomes permanently positive.

Answer 1

The Correct Option is B

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 $m=5$.