Sketch the graph of the following function. list the coordinates of where extrema or points of inflection occur. state where the function is increasing or decreasing as well as where it is concave up or concave down. f left parenthesis x right parenthesisf(x)equals=3 x cubed minus 36 x minus 73x3−36x−7

To find extrema we need find where the first derivative is equal to zero.

We now have to find zeros of this equation:

These are x coordinates of our extrema, we calculate y coordinates simply by plugging in x coordinates in our functions:

Function incrases when it's first derivative is positive. 
Our extrema points are (-2,41) and (2,-55).
This is because the first derivative gives you the slope of the function. 
We need to find where our first derivative is positive.
The first derivative is parabola that has positive coefficient a. Parabolas with positive coefficient a ("happy" parabolas or concave) are negative between zeros and positive everywhere else. So our function is increasing when x<-2 and x>2.
In order to determine where the function is concave up  or down, we have to find the second derivative.

When the second derivative is positive function is concave up, whet it is negative function is concave down.
Our function is concave up when x>0 and it is concave down when x<0.