the function (f(x)) has a domain of ((-infty,infty)) and a second - derivative given by (f(x)=12(x +…

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.

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.

Answer

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)