if the interval (a, ∞) describes all values of x for which the graph of f(x) = 4 / (x² - 6x + 9) is…

Explanation:

Step1: Rewrite the function

First, factor the denominator. $f(x)=\frac{4}{x^{2}-6x + 9}=\frac{4}{(x - 3)^{2}}=4(x - 3)^{-2}$.

Step2: Find the derivative

Use the power - rule for differentiation. If $y = u^{n}$, then $y^\prime=nu^{n - 1}u^\prime$. Here $u=x - 3$, $n=-2$. So $f^\prime(x)=4\times(-2)(x - 3)^{-3}\times1=-\frac{8}{(x - 3)^{3}}$.

Step3: Determine where the function is decreasing

A function $y = f(x)$ is decreasing when $f^\prime(x)<0$. Set $f^\prime(x)<0$, so $-\frac{8}{(x - 3)^{3}}<0$. Multiply both sides by - 1 (and reverse the inequality sign) to get $\frac{8}{(x - 3)^{3}}>0$. For a fraction $\frac{a}{b}>0$ (where $a = 8>0$), we need $b=(x - 3)^{3}>0$. Solving $(x - 3)^{3}>0$ gives $x>3$.

Answer:

3