Question C 03
Consider the function
defined on the whole real line.
Find all zeros of and and determine all local minima, maxima and inflection points of f. Sketch the graph of f. You may use the fact that .
Make sure you understand the problem fully: What is the question asking you to do? Are there specific conditions or constraints that you should take note of? How will you know if your answer is correct from your work only? Can you rephrase the question in your own words in a way that makes sense to you?
If you are stuck, check the hint below. Consider it for a while. Does it give you a new idea on how to approach the problem? If so, try it!
Take derivatives using the product rule.
Then make your sign charts and plot the results.
Checking a solution serves two purposes: helping you if, after having used the hint, you still are stuck on the problem; or if you have solved the problem and would like to check your work.
We begin by finding the zeroes of the original function,
and since always, only is a root. But we note that is always greater or equal to zero.
Next we take the derivative and see that
Finding the roots gives us . Making a sign chart, we see that
Graphically, we have
and thus we have that is a local minimum and is a local maximum. The coordinates are given by and (note that is roughly ).
Lastly taking the second derivative, we have that
The zeroes of the second derivative are given by .
Using the quadratic formula gives us the roots
Making a sign chart, we see that
Graphically we have
Hence, we have that are inflection points. Combining all this information, we have the following. First plot all known points of interest and label where the function is increasing, decreasing, concave up and concave down. Then connect the dots in a consistent way. Don't forget that was given to us and we can use this information.