Question

the function $f(x)$ has a domain of $(-\infty, \infty)$ and a second derivative given by $f(x)=12(x + 6)^7(x - 1)^5$. find the $x$-value(s) of the inflection points of $f(x)$. $x =$ if there is more than one, separate your answers by commas. if there are no inflection points, enter none.

Explanation:

Step1: Recall inflection - point condition

Inflection points occur where \(f''(x) = 0\) or \(f''(x)\) is undefined. Since \(f''(x)=12(x + 6)^{7}(x - 1)^{5}\) is a polynomial, it is defined for all \(x\in(-\infty,\infty)\). So we set \(f''(x)=0\).
$$12(x + 6)^{7}(x - 1)^{5}=0$$

Step2: Solve the equation

Using the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\).
For \((x + 6)^{7}=0\), we get \(x=-6\).
For \((x - 1)^{5}=0\), we get \(x = 1\).

Answer:

\(x=-6,1\)