m is the smallest positive integer such that for any integer n $<$ m, the quantity $n^{3} - 7 n^{2} + 11 n - 5$ is positive. What is the value of m?

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None of these

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Solution

The correct option is D None of these
Option (d)

n=1 is a root of the equation

$(n - 1) (n^{2} - 6 n + 5) = (n - 1)^{2} (n - 5)$

Now, $n - 1^{2}$ is always positive.

Now, for n $<$ 5, the expression gives a negative quantity.

Therefore, the least value of n will be 6. Hence m = 7 .

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