Analysis of a function involving the exponential

General method

To analyse a function of the form f(x) = u(x) * e^(v(x)), follow these steps: determine the domain, calculate f'(x) using the product rule and the fact that (e^v)' = v' * e^v, examine the sign of f'(x) (using the fact that e^v(x) is always greater than 0), then draw up the table of variations.

Complete example

Let f(x) = x * e^(-x) on R. Let’s calculate f’:

f'(x) = 1 * e^(-x) + x * (-1) * e^(-x) = e^(-x) * (1 - x)

Since e^(-x) > 0 for all x, the sign of f'(x) is that of (1 - x):

x	-∞	...	1	...	+∞
f'(x)		+	0	-
f(x)		increasing	max = 1/e	decreasing

f therefore has a maximum at x = 1, with f(1) = 1 * e^(-1) = 1/e.

Behaviour at the limits

Thanks to the limits of comparative growth, lim x → +∞ x * e^(-x) = 0 (the exponential term dominates), and lim x → -∞ x * e^(-x) = -∞.

Common pitfall

When differentiating a product u(x) * e^(v(x)), do not forget the product rule: f' = u' * e^v + u * v' * e^v. Omitting the second term is a very common mistake.

Key points

• e^v(x) never cancels out: it does not change the sign of f'.
• Always use the product rule to differentiate u(x) * e^(v(x)).
• Limits of comparative growth resolve indeterminate forms involving e.