the graph of $g$ is given below:\nanswer the following questions about the graph of $g$ based on the…

the graph of $g$ is given below:\nanswer the following questions about the graph of $g$ based on the information related to $g$ and $g$.\na) state the $x$-values over which $g$ is increasing.\nb) state the $x$-values over which $g$ is decreasing.\nc) state the $x$-values where $g$ has horizontal tangent lines.\nd) state the $x$-values over which $g$ is differentiable.

the graph of $g$ is given below:\nanswer the following questions about the graph of $g$ based on the information related to $g$ and $g$.\na) state the $x$-values over which $g$ is increasing.\nb) state the $x$-values over which $g$ is decreasing.\nc) state the $x$-values where $g$ has horizontal tangent lines.\nd) state the $x$-values over which $g$ is differentiable.

Answer

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))