Don't forget to make your x, y table of values in this problem to properly determine the (x, y) coordinates they request.
Part (a)
Recall the tips I gave you about finding the domain of
a function back in Lesson 1, questions 2 and 3. Make sure you are clear about the domain here!
Do part (c) before you do part (b)!
Part (c)
Since they are asking you to show your work, I suggest that you do the one-sided k/0 limits for all the bottom zeros of f(x) here to prove if y is going up to
infinity or down to negative infinity as it approaches each bottom zero from each side.
Be sure to state the equation of the vertical asymptote properly. For example, if there is a vertical asymptote at 10, don't say "10," say "Vertical Asymptote at x = 10."
Otherwise, the rest of this problem is doing a complete first derivative analysis as I teach in Lesson
9.
Part (b)
Vertical Asymptotes are not critical numbers! Your crtical numbers are the critical points (top zeros of f'(x)) and singular points (bottom zeros of f'(x) that are not already known to be vertical asymptotes).
Parts (d) and (e)
You will use a sign diagram for
f'(x) to identify increase and decrease.
Make sure you state the intervals. Don't use the union symbol "U" if there are two or more intervals of increase/decrease. Some profs object to uniting intervals with the union symbol for some reason.
For example, if you establish f(x) is increasing from -5 to -1, and also from 3 to 10, just write these intervals in a list separated by commas, always using round
brackets.
Thus, f(x) is increasing on (-5, -1), (3, 10).
Part (f)
By "first derivative test," they simply mean, use the sign diagram you made for increase/decrease earlier in the problem to visualize whether the critical points are local mins or max.
Part (g)
I discuss and illustrate the second derivative test in Lesson 9,
Practise Problem 20. Note that you are not asked to do any analysis with the second derivative (such as concavity and inflection points), you are just asked to use the second derivative test to check the critical points. Obviously, you should reach the same conclusion as you did in the previous part.
That hint about the polynomial is basically telling you that part of f" is always a positive amount (and, therefore, it has no zero,
either), which may be helpful since all you need to know when performing the second derivative test is the sign of f" at the critical point. If you prefer, you could make a complete sign diagram for f" and then that will tell you its sign at each critical point. In fact, the concavity symbols for each region will make it easy for you to visualize if the critical point is a local max or local min.