Question
Consider the function below. (If you need to use - or , enter -INFINITY or INFINITY.) (a) Find the vertical and horizontal asymptotes. x = (smaller value) x = (larger value) y = (b) Find the intervals where the function is increasing. (Enter the interval that contains smaller numbers first.) ( , ) ( , ) Find the intervals where the function is decreasing. (Enter the interval that contains smaller numbers first.) ( , ) ( , ) (c) Find the local maximum value. (d) Find the intervals where the function is concave up. (Enter the interval that contains smaller numbers first.) ( , ) ( , ) Find the interval where the function is concave down. ( , )
Explanation / Answer
First find critical points (where f'(x) = 0) f'(x) = 3x^2 - 27 = 0 x^2 = 9 x = 3, -3 The critical points are where a function changes from increasing to decreasing. The intervals between them can be either increasing or decreasing. You can find f'(x) for one value in each of the intervals (-infinity, -3), (-3, 3), and (3, infinity). If f' is positive in that interval, it is increasing. If f' is negative, it's decreasing. f'(-4) > 0 --> increasing on (-infinity, -3) f'(0) < 0 --> decreasing on (-3, 3) f'(4) > 0 --> increasing on (3, infinity)