The number of distinct integers n for which log$\left(\frac{1}{4}\right)(n^2-7n+11)>\mathrm{0\; ,is}$

The correct answer is 2.

We need to find the number of distinct integers n for which:

${\log }_{\frac{1}{4}}(n^2-7n+11)\ge 0$

Step 1 – Recall the behaviour of logarithms with base between 0 and 1.

The base of the logarithm here is $\frac{1}{4}$, which satisfies 0 < base < 1. For such a base, the logarithmic function is decreasing. This means:

${\log }_{\frac{1}{4}}\left(x\right)\ge 0$   if and only if   $0<x\le 1$

(Since ${\log }_{\frac{1}{4}}\left(1\right)=0$, and the function is decreasing, it is non-negative only when the argument is at most 1, while remaining positive for the log to be defined.)

Step 2 ��� Set up the compound inequality.

Letting $x=n^2-7n+11$, we need:

$0<n^2-7n+11\le 1$

Step 3 – Solve the right-hand inequality: n² − 7n + 11 ≤ 1.

$n^2-7n+11\le 1$

$n^2-7n+10\le 0$

Factorising:

$(n-2)(n-5)\le 0$

This quadratic inequality holds when $2\le n\le 5$.

Step 4 – Solve the left-hand inequality: n² − 7n + 11 > 0.

The discriminant of $n^2-7n+11$ is:

$D=49-44=5$

The roots are:

$n=\frac{7\pm \sqrt{5}}{2}$

Since $\sqrt{5}\approx 2.236$, the roots are approximately 2.382 and 4.618.

The quadratic is positive (opens upwards) when $n<2.382$ or $n>4.618$.

Step 5 – Find the intersection of both conditions for integer values of n.

We need integers n satisfying both:

• $2\le n\le 5$  (from Step 3)

• $n<2.382$ or $n>4.618$  (from Step 4)

The integer candidates in [2, 5] are: n = 2, 3, 4, 5.

Now applying the second condition:

• n = 2:   2 < 2.382 ✔  →  n² − 7n + 11 = 4 − 14 + 11 = 1 > 0 ✔

• n = 3:   3 is NOT < 2.382 and NOT > 4.618 ✘  →  n² − 7n + 11 = 9 ��� 21 + 11 = −1 < 0 (argument is negative, log undefined)

• n = 4:   4 is NOT < 2.382 and NOT > 4.618 ✘  →  n² − 7n + 11 = 16 − 28 + 11 = −1 < 0 (argument is negative, log undefined)

• n = 5:   5 > 4.618 ✔  →  n² − 7n + 11 = 25 − 35 + 11 = 1 > 0 ✔

Step 6 – Verify the two valid values.

For n = 2 and n = 5, the argument equals 1:

${\log }_{\frac{1}{4}}\left(1\right)=0\ge 0$   ✔

These are the only integers in the valid range, since n = 3 and n = 4 make the argument negative (−1), rendering the logarithm undefined.

Therefore, the number of distinct integers n satisfying the given inequality is 2.