the function $f(x)$ has a domain of $(-\\infty, \\infty)$ and a second derivative given by $f(x)=-5(x…

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

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

Answer

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)