Verified Answer :
- Factor the denominator: \(f(x) = \frac{4}{(x - 3)^2}\).
- Find the derivative using the power and chain rules: \(f'(x) = -\frac{8}{(x - 3)^3}\).
- Set the inequality \(f'(x) < 0\) to find where the function is decreasing.
- Solve \(-\frac{8}{(x - 3)^3} < 0\) to find \(x > 3\), leading to \(a = \boxed{3}\).
Explanation
- Function Analysis and Simplification
We are analyzing the function \(f(x) = \frac{4}{x^2 - 6x + 9}\). First, notice that the denominator can be factored as a perfect square: \(x^2 - 6x + 9 = (x - 3)^2\). Therefore, the function can be rewritten as \(f(x) = 4(x - 3)^{-2}\). This function is defined for all real numbers except \(x = 3\), where the function has a vertical asymptote.
- Finding the Derivative
To determine where the function is decreasing, we must find the derivative \(f'(x)\) and identify the interval where \(f'(x) < 0\). Using the chain rule, the derivative of \(f(x) = 4(x - 3)^{-2}\) is:
\(f'(x) = 4 \cdot (-2) \cdot (x - 3)^{-3} \cdot 1 = -8(x - 3)^{-3} = -\frac{8}{(x - 3)^3}\).
- Determining the Decreasing Interval
The function \(f(x)\) is decreasing when \(f'(x) < 0\). We set up the inequality:
\(-\frac{8}{(x - 3)^3} < 0\).
Since the numerator \(-8\) is constant and negative, the entire fraction is negative if and only if the denominator \((x - 3)^3\) is positive.
\((x - 3)^3 > 0\)
Taking the cube root of both sides, we get \(x - 3 > 0\), which simplifies to \(x > 3\). Thus, the function is decreasing on the interval \((3, \infty)\).
- Final Answer Identification
Comparing the interval \((3, \infty)\) to the given interval \((a, \infty)\), we identify that \(a = 3\).
Examples
In engineering and physics, understanding the behavior of functions is crucial. For instance, the intensity of light or sound from a point source follows an inverse-square law, often modeled by functions similar to \(f(x) = \frac{k}{x^2}\). Knowing the intervals where such a function is decreasing helps in predicting how quickly the signal strength fades as distance increases, allowing engineers to design more efficient communication or lighting systems.