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)
2≤x<4 negative positive
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 (use a comma to separate answers as needed.)

Explanation:

Step1: Recall extreme - point condition

A function \(y = f(x)\) has a relative extreme point at \(x = c\) when \(f^{\prime}(c)=0\) and the sign of \(f^{\prime}(x)\) changes around \(x = c\).

Step2: Analyze \(f^{\prime}(x)\) values

We see that \(f^{\prime}(4) = 0\). For \(2\leq x<4\), \(f^{\prime}(x)\) is negative and for \(4 < x<5\), \(f^{\prime}(x)\) is positive. So the sign of \(f^{\prime}(x)\) changes from negative to positive at \(x = 4\), indicating a relative minimum. Also, \(f^{\prime}(6)=0\), for \(5 < x<6\), \(f^{\prime}(x)\) is positive and for \(6 < x\leq8\), \(f^{\prime}(x)\) is positive. Since the sign of \(f^{\prime}(x)\) does not change at \(x = 6\), it is not a relative - extreme point.

Answer:

4