Given: f(x)=(x+2)e−x Here we have to find the interval in which f(x) is decreasing. Let's first calculate f′(x) As we know that,
d
dx
[f(x).g(x)]=f(x).
d{g(x)}
dx
+g(x).
d{f(x)}
dx
⇒f′(x)=(x+2).
d(e−x)
dx
+e−x.
d(x+2)
dx
As we know that,
d(eax)
dx
=a.eax ⇒f′(x)=−(x+2).e−x+e−x.1 ⇒f′(x)=−(x+1).e−x As we know that for a decreasing function say f(x) we have f′(x)≤0 ⇒−(x+1).e−x≤0 ⇒(x+1).e−x≥0 ⇒x+1≥0 ........(∵e−x>0) ⇒x≥−1 Hence, the interval in which the given function is decreasing is [−1,∞)