Question

the first and second derivatives of the function f(x) have the values given in the table below. find the x - coordinates of all relative extreme points and find the x - coordinates of all inflection points.

x	f(x)	f(x)
4	0	positive
4<x<5	positive	positive
5	positive	0
5<x<6	positive	negative
6	0	0
6<x≤8	positive	positive

the x - coordinates for the relative extreme points are 4 (use a comma to separate answers as needed.)
the x - coordinates for the inflection points are (use a comma to separate answers as needed.)

Explanation:

Step1: Recall relative - extreme point condition

A relative extreme point occurs where $f^{\prime}(x)=0$ and the sign of $f^{\prime}(x)$ changes around that point. At $x = 4$, $f^{\prime}(4)=0$, and $f^{\prime}(x)$ is negative for $2\leq x<4$ and positive for $4 < x<5$, so $x = 4$ is a relative minimum.

Step2: Recall inflection - point condition

An inflection point occurs where the concavity changes, i.e., where $f^{\prime\prime}(x)$ changes sign. At $x = 5$, $f^{\prime\prime}(x)$ is positive for $4 < x<5$ and negative for $5 < x<6$.

Answer:

The x - coordinates for the relative extreme points are 4.
The x - coordinates for the inflection points are 5.