Question

the graph of $g$ is given below:
answer the following questions about the graph of $g$ based on the information related to $g$ and $g$.
a) state the $x$-values over which $g$ is increasing.
b) state the $x$-values over which $g$ is decreasing.
c) state the $x$-values where $g$ has horizontal tangent lines.
d) state the $x$-values over which $g$ is differentiable.

Explanation:

Step1: Determine when \(g\) is increasing

A function \(g(x)\) is increasing when \(g^{\prime}(x)>0\). Looking at the graph of \(y = g^{\prime}(x)\), we find the intervals where the graph is above the \(x -\)axis.
From the left - hand open circle (assuming the domain starts from the left - most point of the non - dotted part) to the local minimum (where \(g^{\prime}(x) = 0\)) and from the local minimum (where \(g^{\prime}(x)=0\)) after the \(x = 0\) (peak of \(g^{\prime}(x)\)) to the right - hand side (where \(g^{\prime}(x)>0\) again).
If we assume the grid is such that the left - hand open circle is at \(x=-4\), the first local minimum (where \(g^{\prime}(x) = 0\)) is at \(x=-2\), the local maximum of \(g^{\prime}(x)\) is at \(x = 0\), the \(x\) - value where \(g^{\prime}(x)\) crosses the \(x\) - axis (going from positive to negative) is \(x = 2\) and the local minimum of \(g^{\prime}(x)\) (where \(g^{\prime}(x)=0\)) is at \(x=4\).
The function \(g(x)\) is increasing when \(g^{\prime}(x)>0\). So \(g(x)\) is increasing on \((-\infty,-4)\cup(-2,2)\cup(4,\infty)\)

Step2: Determine when \(g\) is decreasing

A function \(g(x)\) is decreasing when \(g^{\prime}(x)<0\). Looking at the graph of \(y = g^{\prime}(x)\), we find the intervals where the graph is below the \(x -\)axis.
The function \(g(x)\) is decreasing on \((-4, - 2)\cup(2,4)\)

Step3: Determine where \(g\) has horizontal tangent lines

A function \(g(x)\) has a horizontal tangent line when \(g^{\prime}(x)=0\). The \(x\) - values of the \(x\) - intercepts (where \(y = g^{\prime}(x)=0\)) and the local minima/maxima of \(g^{\prime}(x)\) (where the slope of \(g^{\prime}(x)\) is \(0\), but for \(g(x)\) the derivative \(g^{\prime}(x)\) exists).
The \(x\) - values are \(x=-2,x = 2,x = 4\)

Step4: Determine where \(g\) is differentiable

A function \(y = g(x)\) is differentiable at all \(x\) - values in its domain where \(g^{\prime}(x)\) exists. There is a hole (open circle) at \(x=-4\). So \(g(x)\) is differentiable on \((-\infty,-4)\cup(-4,\infty)\)

Answer:

A. \(g\) is increasing on \((-\infty,-4)\cup(-2,2)\cup(4,\infty)\)
B. \(g\) is decreasing on \((-4, - 2)\cup(2,4)\)
C. \(g\) has horizontal tangent lines at \(x=-2,x = 2,x = 4\)
D. \(g\) is differentiable on \((-\infty,-4)\cup(-4,\infty)\)