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 $f(x)=12(x + 6)^7(x - 1)^5$. 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.

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.

Answer

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)