A function f is increasing on an interval when
[tex]f^{\prime}(x)>0\text{ for all x in that interval}[/tex]
Also,
[tex]\begin{gathered} f^{\prime}(x)>0\text{ for all x in that interval if the tangents to the curve at any point on the curve } \\ \text{ in that interval makes an acute angle with the positive x-axis} \end{gathered}[/tex]
From the image,
the function has a maximum point at x = -8,
therefore
[tex]\begin{gathered} f^{\prime}(x)>0\text{ for x < -8} \\ \text{ That is } \\ f^{\prime}(x)>0\text{ on (-}\infty,-8) \end{gathered}[/tex]
Also,
[tex]\begin{gathered} f^{\prime}(x)>0,for \\ -30\text{ on (-3, -2)} \end{gathered}[/tex]
Hence, the function is increasing on the intervals
[tex](-\infty,-8)\text{ and (-3, -2)}[/tex]
b)
A function f is decreasing on an interval when
[tex]f^{\prime}(x)<0\text{ for all x in that interval}[/tex]
Also,
[tex]\begin{gathered} f^{\prime}(x)<0\text{ for all x in that interval if the tangents to the curve at any point on the curve } \\ \text{in that interval makes an angle that is not acute with the positive x-axis} \end{gathered}[/tex]
From the image,
the function has a maximum point at x = -8,
therefore
[tex]\begin{gathered} f^{\prime}(x)<0\text{ for }-8Hence, the function is decreasing on the interval[tex](-8,-6)[/tex]
(c)
A function f is constant on an interval when,
[tex]f^{\prime}(x)=0\text{ for all x in that interval}[/tex]
Also,
[tex]f^{\prime}(x)=0\text{ for all x in that interval if the graph is parallel to the x-axis on that interval}[/tex]
From the image, we can see that the graph is parallel to the x-axis on the intervals
[tex]-6-2}[/tex]
Hence, the function is constant on the intervals
[tex](-6,-3)\text{ and (-2, }\infty)[/tex]