suppose that g(x) is a polynomial function, g(6) = 19, g(6) = 0, and g(6) = 10. which of the following is…

Explanation:

Step1: Recall derivative - based function behavior rules

The first - derivative $g^{\prime}(x)$ determines if a function is increasing or decreasing. If $g^{\prime}(x)>0$, the function is increasing; if $g^{\prime}(x)<0$, the function is decreasing. The second - derivative $g^{\prime\prime}(x)$ is used to determine concavity and relative extrema. A critical point occurs where $g^{\prime}(x) = 0$.

Step2: Analyze the first - derivative value

Given that $g^{\prime}(6)=0$, $x = 6$ is a critical point of the function $g(x)$. This means the function has a horizontal tangent at $x = 6$, so it is neither increasing ($g^{\prime}(x)>0$) nor decreasing ($g^{\prime}(x)<0$) at $x = 6$. So, options A and C are incorrect.

Step3: Analyze the second - derivative value

The second - derivative test states that if $g^{\prime}(c)=0$ and $g^{\prime\prime}(c)>0$, then the function $g(x)$ has a relative minimum at $x = c$. Here, $c = 6$, $g^{\prime}(6)=0$, and $g^{\prime\prime}(6)=10>0$. So, $g(x)$ has a relative minimum at $x = 6$.

Answer:

B. $g(x)$ has a relative minimum at $x = 6$