Question

The number of non-negative values of n for which \(\log_{1/4}(n^2 - 7n + 14)>0\) is ____.

Hint:

For logarithmic inequalities with a base between 0 and 1, remember to reverse the inequality sign. Analyzing the minimum or maximum value of a quadratic argument is often the quickest way to solve.

Answer 1

Correct Answer: 0

Step 1: Understanding the Question:
We need to solve a logarithmic inequality. The key features are the base of the logarithm, which is between 0 and 1, and the quadratic expression as the argument.
Step 2: Key Formula or Approach:
For a logarithmic inequality \(\log_{b}(A)>C\) where the base \(b\) is between 0 and 1 (\(0<b<1\)), the inequality sign is reversed when converting to exponential form. Also, the argument A must be positive.
So, \(\log_{b}(A)>C\) becomes \(0<A<b^C\).
Step 3: Detailed Explanation:
In our case, \(b = 1/4\), \(A = n^2 - 7n + 14\), and \(C = 0\).
Applying the rule, the inequality \(\log_{1/4}(n^2 - 7n + 14)>0\) is equivalent to:
\[ 0<n^2 - 7n + 14<(1/4)^0 \] \[ 0<n^2 - 7n + 14<1 \] This gives us two separate inequalities to solve:
1) \(n^2 - 7n + 14>0\)
2) \(n^2 - 7n + 14<1 \implies n^2 - 7n + 13<0\)
Let's analyze the quadratic expression \(f(n) = n^2 - 7n + 14\). It is an upward-opening parabola. Let's find its minimum value by finding the vertex.
The vertex occurs at \(n = -\frac{-7}{2(1)} = 3.5\).
The minimum value of the expression is \(f(3.5) = (3.5)^2 - 7(3.5) + 14 = 12.25 - 24.5 + 14 = 1.75\).
Since the minimum value of \(n^2 - 7n + 14\) is 1.75, the expression is always greater than 0. So, the first inequality is always true.
However, for the second inequality, \(n^2 - 7n + 14<1\), since the minimum value of the left side is 1.75, it can never be less than 1. Thus, the second inequality has no solution.
Since there are no real values of n that satisfy the system, there are no non-negative values either.
Step 4: Final Answer:
The number of non-negative values of n is 0.