I am attaching a picture of the problem, but first some background information.

I know that the graph is increasing if the first derivative is above the x-axis and decreasing if the first derivative is below the x-axis. That said I would like to make sure my intervals are correct for part a.

Second, I know that the graph of f must have a local min or max if the derivative graph is 0 and it is a min if it goes from neg to pos, and a max if it goes from pos to neg. Otherwise if it is 0 and does not cross the x-axis it is a saddle point (Like x = -2 is).

Third, I know that f is concave up if the SECOND derivative is pos and concave down if the SECOND derivative is neg. This is where I am stuck because I don't remember (and more importantly cant find a good example with this first derivative graph) on what to do. I do know that if the tangent line is zero on the first derivative then it is an inflection point, but I don't seem to understand how to determine if it is concave up or down on the original function. (There are two inflection points x = -2 and x = 0)

Finally, can anyone tell me what is happening on x = 2 for the original graph because I do not know how I utilize that point discontinuity to draw a continuous graph. (Unless does it mean that the slope is vertical thus undefined and it changes from increasing to decreasing)?

Thanks and here is the picture

< link to objectionable page removed >

I know that the graph is increasing if the first derivative is above the x-axis and decreasing if the first derivative is below the x-axis. That said I would like to make sure my intervals are correct for part a.

Second, I know that the graph of f must have a local min or max if the derivative graph is 0 and it is a min if it goes from neg to pos, and a max if it goes from pos to neg. Otherwise if it is 0 and does not cross the x-axis it is a saddle point (Like x = -2 is).

Third, I know that f is concave up if the SECOND derivative is pos and concave down if the SECOND derivative is neg. This is where I am stuck because I don't remember (and more importantly cant find a good example with this first derivative graph) on what to do. I do know that if the tangent line is zero on the first derivative then it is an inflection point, but I don't seem to understand how to determine if it is concave up or down on the original function. (There are two inflection points x = -2 and x = 0)

Finally, can anyone tell me what is happening on x = 2 for the original graph because I do not know how I utilize that point discontinuity to draw a continuous graph. (Unless does it mean that the slope is vertical thus undefined and it changes from increasing to decreasing)?

Thanks and here is the picture

< link to objectionable page removed >

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