Calculus I problem

Question

Answer the following questions about the function whose derivative is f'(x)=(x-1)e^-x a. What are the critical points of f? b. On what open intervals is f increasing or decreasing? c. At what points, if any, does f assume local maximum and minimum values?

Answer 1

You could do an integration by parts or simply observe the obvious: f(x) = -x e^-x + C

There is only one critical point, x = 1, as (x-1)e^-x = 0 at x = 1.

f(x) is decreasing on (-∞,1) and increasing on (1,∞), as e^-x is always positive, while x-1 is positive for x > 1 and negative for x < 1.

The local and global minimum is at x = 1.There is no maximum. Note that there is a right asymptote equal to C. As x goes to -∞, the function goes to ∞.