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Question:

m is the smallest positive integer such that for any integer n > m, the quantity n³ – 7n² + 11n – 5 is positive. What is the value of m?

Explanation:

n³ − 7n² + 11n − 5

For n = 1, the expression reduces to zero

∴ (n − 1) is a factor.

n³ − 7n² + 11n − 5 = n³ − n² − 6n² + 6n + 5n − 5

                               = n²(n − 1) − 6n(n − 1) + 5(n − 1)

                               = (n − 1)(n² − 6n + 5)

                               = (n − 1)(n − 1)(n − 5)

                               = (n − 1)²(n − 5)

Since, (n − 1)² is always positive.

∴ The expression is positive only when n > 5.

∴ n = 6 and m = 5

Hence, option (b).

Alternatively,

Check using options. If we take m = 4 then n = 5, then we do not get a positive value of the given equation. Therefore, check the next higher value.

n = 6 and m = 5 gives the positive value.

Hence, option (b).