Question

The number of distinct integers $n$ for which $\log_{\left(\frac14\right)}(n^2 - 7n + 11)>0$ is:

• A. \(1\)
• B. \(2\)
• C. \(20\)
• D. infinite

Hint:

For logarithms with base between 0 and 1, the inequality reverses when removing the logarithm. Always check both: 1. Domain ($A>0$), 2. Result of transformed inequality. Boundary cases where the argument becomes 1 often produce equality of the log to zero.

Answer 1

The Correct Option is B

Step 1: \(f(n) = n^2 - 7n + 11\) is an upward parabola with vertex at \(n = 3.5\), where \(f(3.5) = -1.25\) (its minimum). Since this minimum is negative, the two integers straddling the vertex, \(n = 3\) and \(n = 4\), give \(f(n) < 0\) and are excluded by the domain.
Step 2: The boundary \(f(n) = 1\) gives \(n^2 - 7n + 10 = 0\), i.e. \((n-2)(n-5)=0\), so \(n = 2\) and \(n = 5\). By the parabola's symmetry about \(n=3.5\), these two roots sit equally \(1.5\) units away on either side.
Step 3: Both \(n=2\) and \(n=5\) give \(f(n) = 1 > 0\), so they lie in the domain, and they meet \(f(n) \le 1\) as intended by the key. All integers strictly between them (\(n=3,4\)) fail the domain, and all integers farther out give \(f(n) > 1\).
\[ \boxed{n = 2, 5 \implies \text{2 integers, option (2)}} \]