Question

the function f(x) has a domain of (-∞, ∞) and a second - derivative given by f(x)=12(x + 10)^11(x - 2)^5. 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 \(f''(x)=12(x + 10)^{11}(x - 2)^{5}\).

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

Set \(12(x + 10)^{11}(x - 2)^{5}=0\). By the zero - product property, if \(ab = 0\), then \(a = 0\) or \(b = 0\). So \((x + 10)^{11}=0\) gives \(x=-10\) and \((x - 2)^{5}=0\) gives \(x = 2\).

Step3: Check for undefined points

Since \(f''(x)\) is a polynomial (a product of polynomial factors), it is defined for all \(x\in(-\infty,\infty)\).

Answer:

\(-10,2\)