Question

the function $f(x)$ has a domain of $(-\infty, \infty)$ and a second derivative given by $f(x)=-5(x - 1)^3(x + 7)^5$. 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. next item

Explanation:

Step1: Recall inflection - point condition

Inflection points occur where \(f''(x) = 0\) or \(f''(x)\) is undefined. Since \(f''(x)=-5(x - 1)^{3}(x + 7)^{5}\) is a polynomial, it is defined for all real \(x\). So we set \(f''(x)=0\).
\(-5(x - 1)^{3}(x + 7)^{5}=0\)

Step2: Solve the equation

Using the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\).
For \((x - 1)^{3}=0\), we get \(x=1\). For \((x + 7)^{5}=0\), we get \(x=-7\).

Answer:

\(x = 1,-7\)