Find the intervals in which f(x) = (x + 2)e^{–x} is increasing or decreasing ?

we have,

f(x) = (x + 2)e^{–x}

f'(x) = e^{–x} – e^{–x} (x+2)

= e^{–x} (1 – x – 2)

= – e^{–x} (x+1)

Critical points

f'(x) = 0

⇒ – e^{–x} (x + 1) = 0

⇒x = – 1

Clearly

Hence f(x) increases in (–∞,–1), decreases in (–1, ∞)

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