Question #6c313
1 Answer
Please see below.
Explanation:
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A function is monotonic if it is either entirely non-increasing or non-decreasing. To find out, we need to take the derivative of it:
We use the Quotient Rule to take its derivative:
Let's set this derivative equal to zero and solve for its roots:
These are the critical points of the function. If we evaluate the derivative function at values to the left and right of each root we find that at
This indicates that at
At
Setting the function equal to zero and solving for its roots gives us
Furthermore, at
This indicates that at
As a result of the fact that the derivative of the function changes signs, the function is not monotone.
To find the asymptotes of the function we have to consider that it is a rational function. Therefore, we set the denominator equal to zero and solve for its roots:
We also notice that the numerator is of degree
To find it, we do a long division to divide the numerator by the denominator which gives us:
The slant asymptote has the equation:
Here is the graph of the function where you can see the critical points, asymptotes, and the behavior of the function as described above: