Question

the function (f(x)) has a domain of ((-infty,infty)) and a second - derivative given by (f(x)=12(x + 7)^{11}(x - 2)^{7}). find the (x) - value(s) of the inflection points of (f(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. Given \(f''(x)=12(x + 7)^{11}(x - 2)^{7}\).

Step2: Set \(f''(x)=0\)

Set \(12(x + 7)^{11}(x - 2)^{7}=0\). By the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\).
For \((x + 7)^{11}=0\), we solve \(x+7 = 0\), getting \(x=-7\).
For \((x - 2)^{7}=0\), we solve \(x - 2=0\), getting \(x = 2\).
Since \(f''(x)\) is a polynomial, it is defined for all real \(x\) (because the domain of \(f(x)\) is \((-\infty,\infty)\)).

Answer:

\(-7,2\)