To draw the graph of a function, follow these steps:
- Domain and continuity
- Find f'(x) and determine increase and decrease
- Find f''(x) and determine concavity
- Limits (asymptotes)
- Intercepts and symmetry
Example ① Investigate the general shape of y = x·e^{−x}.
Solution: y' = (1−x)e^{−x}. It is increasing when x<1 and decreasing when x>1. At x=1, it has a local maximum value of 1/e.
y'' = (x−2)e^{−x}. It is concave downward when x<2 and concave upward when x>2. The inflection point is (2, 2/e²).
As x→∞, y→0; as x→−∞, y→−∞.
Solution: y' = (1−x)e^{−x}. It is increasing when x<1 and decreasing when x>1. At x=1, it has a local maximum value of 1/e.
y'' = (x−2)e^{−x}. It is concave downward when x<2 and concave upward when x>2. The inflection point is (2, 2/e²).
As x→∞, y→0; as x→−∞, y→−∞.
Key Point
- Create the variation and concavity tables together
- Do not forget horizontal and vertical asymptotes
Practice Problem
- Find the local extrema of y = log x / x.
Answer and Explanation
- Answer:Local maximum value 1/e at x = e
y' = (1−log x)/x².