So far although we have said that critical points of a function are the endpoints and points at which the derivative is either zero or undefined, we have had very few, if any, examples in which f' is undefined at a point in the domain of f. The first problem below (along with the ones to be assigned on Friday) are an attempt to rectify this!
 
 

1.

Let h(x) = |g(x)|, where g(x) = x² -1.

(a)

Graph g(x) and h(x) on the same set of axes.

(b)

What are the critical points of g?

What are the critical points of h?

(c)

Identify all local extrema of h.

(d)

Does h have an absolute maximum value? If so, what is it, and where is it attained?

(e)

Does h have an absolute minimum value? If so, what is it, and where is it attained?

2.

Let . While you should feel free to use your graphing calculator, the work for this question must stand on its own, completely independent of the machine with the exception of the very last part of the question.

(a)

Identify any x-intercepts, y-intercepts, vertical and horizontal asymptotes of f.

(b)

Find f '(x). Show your work.

(c)

What are the critical points of f? Give exact answers, not numerical approximations from your calculator.

(d)

Graph f(x), labelling the x coordinates of any turning points.

(Note: If there is a critical point at x=a and you know it is a local minimum due to your knowledge of the behavior of a rational function, you need not prove that it is a local minimum. )

(e)

What is the domain of f?

(f)

Approximately what is the range of f? (Here you can use your graphing calculator, in either its graphing or numerical capabilities.)

Return to MathXa Course Page