# Find: For the function below, identify the domain; find the critical points and possible...

## Question:

For the function below, identify the domain; find the critical points and possible inflection points; finder intervals on which the function is increasing/decreasing and concave up/down; identify extreme values; and graph the function.

{eq}g(x)=\frac{x^2+7x+12}{(x-3)^2 (x+1)} {/eq}

What I'm needing to do is graph the above function by hand. I need to determine the critical values of the function. I also need to determine the inflection points of the function. After the critical value is found, I need to determine the intervals and the increases and decreases on each interval. Once the inflection points are found, the concavity needs to be determined for each interval created from the inflection points. I need both the x and y values of the absolute extremum. Basically, the problem is requesting the aforementioned information, so that the graph of the function can be accurately produced based on that information.

## Studying functions with calculus

A quotient polynomic function with one independent variable has a decreasing interval if its first derivative is negative, and increasing interval if its first derivative is positive, also, a function is concave down if its second derivative is negative.

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The function is:

{eq}\displaystyle \ g(x) = \frac{x^2+7x+12}{(x-3)^2 (x+1)} \\ \displaystyle \ g(x)= {\frac { \left( x+4 \right) \left( x+3...