Question

Determine the intervals on which the function is increasing, decreasing, and constant.List the interval(s) on which the function is increasing. [____]List the interval(s) on which the function is decreasing. [____]List the interval(s) on which the function is constant. [____]

Answer 1

A function f is increasing on an interval when

$f^{\prime}(x)>0\text{ for all x in that interval}$

Also,

$\begin{gathered} f^{\prime}(x)>0\text{ for all x in that interval if the tangents to the curve at any point on the curve } \\ \text{ in that interval makes an acute angle with the positive x-axis} \end{gathered}$

From the image,

the function has a maximum point at x = -8,

therefore

$\begin{gathered} f^{\prime}(x)>0\text{ for x < -8} \\ \text{ That is } \\ f^{\prime}(x)>0\text{ on (-}\infty,-8) \end{gathered}$

Also,

$\begin{gathered} f^{\prime}(x)>0,for \\ -30\text{ on (-3, -2)} \end{gathered}$

Hence, the function is increasing on the intervals

$(-\infty,-8)\text{ and (-3, -2)}$

b)

A function f is decreasing on an interval when

$f^{\prime}(x)<0\text{ for all x in that interval}$

Also,

$\begin{gathered} f^{\prime}(x)<0\text{ for all x in that interval if the tangents to the curve at any point on the curve } \\ \text{in that interval makes an angle that is not acute with the positive x-axis} \end{gathered}$

From the image,

the function has a maximum point at x = -8,

therefore

$\begin{gathered} f^{\prime}(x)<0\text{ for }-8Hence, the function is decreasing on the interval[tex](-8,-6)$

(c)

A function f is constant on an interval when,

$f^{\prime}(x)=0\text{ for all x in that interval}$

Also,

$f^{\prime}(x)=0\text{ for all x in that interval if the graph is parallel to the x-axis on that interval}$

From the image, we can see that the graph is parallel to the x-axis on the intervals

$-6-2}$

Hence, the function is constant on the intervals

$(-6,-3)\text{ and (-2, }\infty)$